Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime...

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Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series Tamar Ziegler Patterns in primes

Transcript of Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime...

Page 1: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Dynamics and prime solutions to linear equations

Tamar Ziegler

Technion/Hebrew University

EMS Lecture Series

Tamar Ziegler Patterns in primes

Page 2: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Denote by P the set of prime numbers P = {2, 3, 5, . . .}.

Let {ψi (~x)}ki=1 be k affine linear forms in n variables

ψi (~x) = ~ai · ~x + bi , ~ai ∈ Zn, bi ∈ Z

Question:

Are there ~x ∈ Zn such ψ1(~x), . . . , ψk(~x) ∈ P ?

Infinitely many such ~x? Asymptotics?

Tamar Ziegler Patterns in primes

Page 3: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Denote by P the set of prime numbers P = {2, 3, 5, . . .}.

Let {ψi (~x)}ki=1 be k affine linear forms in n variables

ψi (~x) = ~ai · ~x + bi , ~ai ∈ Zn, bi ∈ Z

Question:

Are there ~x ∈ Zn such ψ1(~x), . . . , ψk(~x) ∈ P ?

Infinitely many such ~x? Asymptotics?

Tamar Ziegler Patterns in primes

Page 4: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Denote by P the set of prime numbers P = {2, 3, 5, . . .}.

Let {ψi (~x)}ki=1 be k affine linear forms in n variables

ψi (~x) = ~ai · ~x + bi , ~ai ∈ Zn, bi ∈ Z

Question:

Are there ~x ∈ Zn such ψ1(~x), . . . , ψk(~x) ∈ P ?

Infinitely many such ~x? Asymptotics?

Tamar Ziegler Patterns in primes

Page 5: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

ψ(x) = ax + b

k = n = 1: If a = 1, b = 0, then

ψ(x) = x .

Theorem (Euclid (∼ 300 BC) )

There are ∞ many primes.

Over 2000 years later ...

Theorem (Hadamard, de la Vallee-Poussin (1896))

π(N) = |{x ∈ P, x ≤ N}| ∼ N

logN

Tamar Ziegler Patterns in primes

Page 6: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

ψ(x) = ax + b

k = n = 1: If a = 1, b = 0, then

ψ(x) = x .

Theorem (Euclid (∼ 300 BC) )

There are ∞ many primes.

Over 2000 years later ...

Theorem (Hadamard, de la Vallee-Poussin (1896))

π(N) = |{x ∈ P, x ≤ N}| ∼ N

logN

Tamar Ziegler Patterns in primes

Page 7: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

ψ(x) = ax + b

k = n = 1: If a = 1, b = 0, then

ψ(x) = x .

Theorem (Euclid (∼ 300 BC) )

There are ∞ many primes.

Over 2000 years later ...

Theorem (Hadamard, de la Vallee-Poussin (1896))

π(N) = |{x ∈ P, x ≤ N}| ∼ N

logN

Tamar Ziegler Patterns in primes

Page 8: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

More generally:ψ(x) = ax + b

Theorem (Johann Peter Gustav Lejeune Dirichlet (1837))

There are ∞ many primes of the form ax + b⇐⇒ (a, b) = 1 (no local obstructions)

Prime number theorem in arithemtic progression

Each legal arithmetic progression gets its fair share: if (a, b) = 1,then

π(N, a, b) = |{x ∈ P, x ≤ N, x ≡ b mod a}| ∼ 1

φ(a)

N

logN

Tamar Ziegler Patterns in primes

Page 9: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

More generally:ψ(x) = ax + b

Theorem (Johann Peter Gustav Lejeune Dirichlet (1837))

There are ∞ many primes of the form ax + b⇐⇒ (a, b) = 1 (no local obstructions)

Prime number theorem in arithemtic progression

Each legal arithmetic progression gets its fair share: if (a, b) = 1,then

π(N, a, b) = |{x ∈ P, x ≤ N, x ≡ b mod a}| ∼ 1

φ(a)

N

logN

Tamar Ziegler Patterns in primes

Page 10: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

More generally:ψ(x) = ax + b

Theorem (Johann Peter Gustav Lejeune Dirichlet (1837))

There are ∞ many primes of the form ax + b⇐⇒ (a, b) = 1 (no local obstructions)

Prime number theorem in arithemtic progression

Each legal arithmetic progression gets its fair share: if (a, b) = 1,then

π(N, a, b) = |{x ∈ P, x ≤ N, x ≡ b mod a}| ∼ 1

φ(a)

N

logN

Tamar Ziegler Patterns in primes

Page 11: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

k = 2, n = 1

ψ1(x) = x ψ2(x) = x + 2

This is the ”twin prime conjecture” !

News Flash: Peter Woit’s blog May 12: Yitang Zhang proved

ψ1(x) = x ψ2(x) = x + M M < 70, 000, 000 !!!

Tamar Ziegler Patterns in primes

Page 12: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

k = 2, n = 1

ψ1(x) = x ψ2(x) = x + 2

This is the ”twin prime conjecture” !

News Flash: Peter Woit’s blog May 12: Yitang Zhang proved

ψ1(x) = x ψ2(x) = x + M M < 70, 000, 000 !!!

Tamar Ziegler Patterns in primes

Page 13: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

k = 2, n = 1

ψ1(x) = x ψ2(x) = x + 2

This is the ”twin prime conjecture” !

Nothing to report on this ...

News Flash: Peter Woit’s blog May 12: Yitang Zhang proved

ψ1(x) = x ψ2(x) = x + M M < 70, 000, 000 !!!

Tamar Ziegler Patterns in primes

Page 14: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

k = 2, n = 1

ψ1(x) = x ψ2(x) = x + 2

This is the ”twin prime conjecture” !

Nothing to report on this ...

News Flash: Peter Woit’s blog May 12: Yitang Zhang proved

ψ1(x) = x ψ2(x) = x + M M < 70, 000, 000 !!!

Tamar Ziegler Patterns in primes

Page 15: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

k = 2, n = 1

ψ1(x) = x ψ2(x) = x + 2

This is the ”twin prime conjecture” !

Nothing to report on this ...News Flash: Peter Woit’s blog May 12: Yitang Zhang proved

ψ1(x) = x ψ2(x) = x + M M < 70, 000, 000 !!!

Tamar Ziegler Patterns in primes

Page 16: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

arithmetic progressions

n = 2, k arbitrary

ψ1(x) = x , ψ2(x) = x + d , . . . , ψk(x) = x + (k − 1)d

This is an k-term arithmetic progression

Theorem (Green-Tao (2004))

The primes contain arbitrarily long arithmetic progressions.

The proof gives a lower bound of the correct order of magnitude(C N2

(logN)k).

Tamar Ziegler Patterns in primes

Page 17: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

arithmetic progressions

n = 2, k arbitrary

ψ1(x) = x , ψ2(x) = x + d , . . . , ψk(x) = x + (k − 1)d

This is an k-term arithmetic progression

Theorem (Green-Tao (2004))

The primes contain arbitrarily long arithmetic progressions.

The proof gives a lower bound of the correct order of magnitude(C N2

(logN)k).

Tamar Ziegler Patterns in primes

Page 18: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

In 2006 Green-Tao proved the following conditional theorem:

Conditional Multidimentional Dirichlet

Let {ψi (~x)}ki=1 be k affine linear integer forms in n variablesSuppose no 2 forms are affinely dependent. Then

{ψi (~x)}ki=1 ⊂ P infinitely often ⇐⇒ No local obstructions

∣∣∣{~x ∈ [0,N]n, {ψi (~x)}ki=1 ⊂ P}∣∣∣ ∼ S(~ψ)

Nn

(logN)k

Conditioned on 2 conjectures: MN(s), and GI (s).

No local obstructions:

mod p: for any prime p, there exists ~x ∈ Zn such that ψi (~x) iscoprime to p all i .

mod ∞: there exist infinitely many ~x ∈ Zn such that ψi (~x) ispositive.

Tamar Ziegler Patterns in primes

Page 19: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

In 2006 Green-Tao proved the following conditional theorem:

Conditional Multidimentional Dirichlet

Let {ψi (~x)}ki=1 be k affine linear integer forms in n variablesSuppose no 2 forms are affinely dependent. Then

{ψi (~x)}ki=1 ⊂ P infinitely often ⇐⇒ No local obstructions

∣∣∣{~x ∈ [0,N]n, {ψi (~x)}ki=1 ⊂ P}∣∣∣ ∼ S(~ψ)

Nn

(logN)k

Conditioned on 2 conjectures: MN(s), and GI (s).

No local obstructions:

mod p: for any prime p, there exists ~x ∈ Zn such that ψi (~x) iscoprime to p all i .

mod ∞: there exist infinitely many ~x ∈ Zn such that ψi (~x) ispositive.

Tamar Ziegler Patterns in primes

Page 20: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

In 2006 Green-Tao proved the following conditional theorem:

Conditional Multidimentional Dirichlet

Let {ψi (~x)}ki=1 be k affine linear integer forms in n variablesSuppose no 2 forms are affinely dependent. Then

{ψi (~x)}ki=1 ⊂ P infinitely often ⇐⇒ No local obstructions

∣∣∣{~x ∈ [0,N]n, {ψi (~x)}ki=1 ⊂ P}∣∣∣ ∼ S(~ψ)

Nn

(logN)k

Conditioned on 2 conjectures: MN(s), and GI (s).

No local obstructions:

mod p: for any prime p, there exists ~x ∈ Zn such that ψi (~x) iscoprime to p all i .

mod ∞: there exist infinitely many ~x ∈ Zn such that ψi (~x) ispositive.

Tamar Ziegler Patterns in primes

Page 21: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

In 2006 Green-Tao proved the following conditional theorem:

Conditional Multidimentional Dirichlet

Let {ψi (~x)}ki=1 be k affine linear integer forms in n variablesSuppose no 2 forms are affinely dependent. Then

{ψi (~x)}ki=1 ⊂ P infinitely often ⇐⇒ No local obstructions

∣∣∣{~x ∈ [0,N]n, {ψi (~x)}ki=1 ⊂ P}∣∣∣ ∼ S(~ψ)

Nn

(logN)k

Conditioned on 2 conjectures: MN(s), and GI (s).

No local obstructions:

mod p: for any prime p, there exists ~x ∈ Zn such that ψi (~x) iscoprime to p all i .

mod ∞: there exist infinitely many ~x ∈ Zn such that ψi (~x) ispositive.

Tamar Ziegler Patterns in primes

Page 22: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

In 2006 Green-Tao proved the following conditional theorem:

Conditional Multidimentional Dirichlet

Let {ψi (~x)}ki=1 be k affine linear integer forms in n variablesSuppose no 2 forms are affinely dependent. Then

{ψi (~x)}ki=1 ⊂ P infinitely often ⇐⇒ No local obstructions

∣∣∣{~x ∈ [0,N]n, {ψi (~x)}ki=1 ⊂ P}∣∣∣ ∼ S(~ψ)

Nn

(logN)k

Conditioned on 2 conjectures: MN(s), and GI (s).

No local obstructions:

mod p: for any prime p, there exists ~x ∈ Zn such that ψi (~x) iscoprime to p all i .

mod ∞: there exist infinitely many ~x ∈ Zn such that ψi (~x) ispositive.

Tamar Ziegler Patterns in primes

Page 23: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

In 2006 Green-Tao proved the following conditional theorem:

Conditional Multidimentional Dirichlet

Let {ψi (~x)}ki=1 be k affine linear integer forms in n variablesSuppose no 2 forms are affinely dependent. Then

{ψi (~x)}ki=1 ⊂ P infinitely often ⇐⇒ No local obstructions

∣∣∣{~x ∈ [0,N]n, {ψi (~x)}ki=1 ⊂ P}∣∣∣ ∼ S(~ψ)

Nn

(logN)k

Conditioned on 2 conjectures: MN(s), and GI (s).

No local obstructions:

mod p: for any prime p, there exists ~x ∈ Zn such that ψi (~x) iscoprime to p all i .

mod ∞: there exist infinitely many ~x ∈ Zn such that ψi (~x) ispositive.

Tamar Ziegler Patterns in primes

Page 24: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

In 2006 Green-Tao proved the following conditional theorem:

Conditional Multidimentional Dirichlet

Let {ψi (~x)}ki=1 be k affine linear integer forms in n variablesSuppose no 2 forms are affinely dependent. Then

{ψi (~x)}ki=1 ⊂ P infinitely often ⇐⇒ No local obstructions

∣∣∣{~x ∈ [0,N]n, {ψi (~x)}ki=1 ⊂ P}∣∣∣ ∼ S(~ψ)

Nn

(logN)k

Conditioned on 2 conjectures: MN(s), and GI (s).

No local obstructions:

mod p: for any prime p, there exists ~x ∈ Zn such that ψi (~x) iscoprime to p all i .

mod ∞: there exist infinitely many ~x ∈ Zn such that ψi (~x) ispositive.

Tamar Ziegler Patterns in primes

Page 25: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Remark: In solving linear equations in primes variables, the nameof the game is

# equations versus # variables

The more variables the easier the question is.

1 equation 2 variables, e.g x1 − x2 = 2. Twin primes.

1 equation 3 variables, e.g the Ternary Goldbach Theorem:every odd number > 5 is a sum of 3 primes: proved byVinogradov (1937) for large enough numbers, very (very)recently proved by Helfgott (13 May 2013) ! or x1 + x3 = 2x2(3 term progressions) proved by Van der Corput (1939). Bothproved using the Hardy Littlewood Circle Method.

Some special cases of k equations f (k) variables - Balog(1992) using the HL circle method.

2 equation 4 variables, e.g x1 + x3 = 2x2; x2 + x4 = 2x3. Thisis 4 term progressions - Green-Tao (2006).

k equation n ≥ k + 2 variables - Conditional MultidimentionalDirichlet. Best possible dependence excluding ”twin primecase”.

Tamar Ziegler Patterns in primes

Page 26: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Remark: In solving linear equations in primes variables, the nameof the game is

# equations versus # variables

The more variables the easier the question is.

1 equation 2 variables, e.g x1 − x2 = 2. Twin primes.

1 equation 3 variables, e.g the Ternary Goldbach Theorem:every odd number > 5 is a sum of 3 primes: proved byVinogradov (1937) for large enough numbers, very (very)recently proved by Helfgott (13 May 2013) ! or x1 + x3 = 2x2(3 term progressions) proved by Van der Corput (1939). Bothproved using the Hardy Littlewood Circle Method.

Some special cases of k equations f (k) variables - Balog(1992) using the HL circle method.

2 equation 4 variables, e.g x1 + x3 = 2x2; x2 + x4 = 2x3. Thisis 4 term progressions - Green-Tao (2006).

k equation n ≥ k + 2 variables - Conditional MultidimentionalDirichlet. Best possible dependence excluding ”twin primecase”.

Tamar Ziegler Patterns in primes

Page 27: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Remark: In solving linear equations in primes variables, the nameof the game is

# equations versus # variables

The more variables the easier the question is.

1 equation 2 variables, e.g x1 − x2 = 2. Twin primes.

1 equation 3 variables, e.g the Ternary Goldbach Theorem:every odd number > 5 is a sum of 3 primes: proved byVinogradov (1937) for large enough numbers, very (very)recently proved by Helfgott (13 May 2013) ! or x1 + x3 = 2x2(3 term progressions) proved by Van der Corput (1939). Bothproved using the Hardy Littlewood Circle Method.

Some special cases of k equations f (k) variables - Balog(1992) using the HL circle method.

2 equation 4 variables, e.g x1 + x3 = 2x2; x2 + x4 = 2x3. Thisis 4 term progressions - Green-Tao (2006).

k equation n ≥ k + 2 variables - Conditional MultidimentionalDirichlet. Best possible dependence excluding ”twin primecase”.

Tamar Ziegler Patterns in primes

Page 28: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Remark: In solving linear equations in primes variables, the nameof the game is

# equations versus # variables

The more variables the easier the question is.

1 equation 2 variables, e.g x1 − x2 = 2. Twin primes.

1 equation 3 variables, e.g the Ternary Goldbach Theorem:every odd number > 5 is a sum of 3 primes:

proved byVinogradov (1937) for large enough numbers, very (very)recently proved by Helfgott (13 May 2013) ! or x1 + x3 = 2x2(3 term progressions) proved by Van der Corput (1939). Bothproved using the Hardy Littlewood Circle Method.

Some special cases of k equations f (k) variables - Balog(1992) using the HL circle method.

2 equation 4 variables, e.g x1 + x3 = 2x2; x2 + x4 = 2x3. Thisis 4 term progressions - Green-Tao (2006).

k equation n ≥ k + 2 variables - Conditional MultidimentionalDirichlet. Best possible dependence excluding ”twin primecase”.

Tamar Ziegler Patterns in primes

Page 29: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Remark: In solving linear equations in primes variables, the nameof the game is

# equations versus # variables

The more variables the easier the question is.

1 equation 2 variables, e.g x1 − x2 = 2. Twin primes.

1 equation 3 variables, e.g the Ternary Goldbach Theorem:every odd number > 5 is a sum of 3 primes: proved byVinogradov (1937) for large enough numbers,

very (very)recently proved by Helfgott (13 May 2013) ! or x1 + x3 = 2x2(3 term progressions) proved by Van der Corput (1939). Bothproved using the Hardy Littlewood Circle Method.

Some special cases of k equations f (k) variables - Balog(1992) using the HL circle method.

2 equation 4 variables, e.g x1 + x3 = 2x2; x2 + x4 = 2x3. Thisis 4 term progressions - Green-Tao (2006).

k equation n ≥ k + 2 variables - Conditional MultidimentionalDirichlet. Best possible dependence excluding ”twin primecase”.

Tamar Ziegler Patterns in primes

Page 30: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Remark: In solving linear equations in primes variables, the nameof the game is

# equations versus # variables

The more variables the easier the question is.

1 equation 2 variables, e.g x1 − x2 = 2. Twin primes.

1 equation 3 variables, e.g the Ternary Goldbach Theorem:every odd number > 5 is a sum of 3 primes: proved byVinogradov (1937) for large enough numbers, very (very)recently proved by Helfgott (13 May 2013) !

or x1 + x3 = 2x2(3 term progressions) proved by Van der Corput (1939). Bothproved using the Hardy Littlewood Circle Method.

Some special cases of k equations f (k) variables - Balog(1992) using the HL circle method.

2 equation 4 variables, e.g x1 + x3 = 2x2; x2 + x4 = 2x3. Thisis 4 term progressions - Green-Tao (2006).

k equation n ≥ k + 2 variables - Conditional MultidimentionalDirichlet. Best possible dependence excluding ”twin primecase”.

Tamar Ziegler Patterns in primes

Page 31: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Remark: In solving linear equations in primes variables, the nameof the game is

# equations versus # variables

The more variables the easier the question is.

1 equation 2 variables, e.g x1 − x2 = 2. Twin primes.

1 equation 3 variables, e.g the Ternary Goldbach Theorem:every odd number > 5 is a sum of 3 primes: proved byVinogradov (1937) for large enough numbers, very (very)recently proved by Helfgott (13 May 2013) ! or x1 + x3 = 2x2(3 term progressions) proved by Van der Corput (1939). Bothproved using the Hardy Littlewood Circle Method.

Some special cases of k equations f (k) variables - Balog(1992) using the HL circle method.

2 equation 4 variables, e.g x1 + x3 = 2x2; x2 + x4 = 2x3. Thisis 4 term progressions - Green-Tao (2006).

k equation n ≥ k + 2 variables - Conditional MultidimentionalDirichlet. Best possible dependence excluding ”twin primecase”.

Tamar Ziegler Patterns in primes

Page 32: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Remark: In solving linear equations in primes variables, the nameof the game is

# equations versus # variables

The more variables the easier the question is.

1 equation 2 variables, e.g x1 − x2 = 2. Twin primes.

1 equation 3 variables, e.g the Ternary Goldbach Theorem:every odd number > 5 is a sum of 3 primes: proved byVinogradov (1937) for large enough numbers, very (very)recently proved by Helfgott (13 May 2013) ! or x1 + x3 = 2x2(3 term progressions) proved by Van der Corput (1939). Bothproved using the Hardy Littlewood Circle Method.

Some special cases of k equations f (k) variables - Balog(1992) using the HL circle method.

2 equation 4 variables, e.g x1 + x3 = 2x2; x2 + x4 = 2x3. Thisis 4 term progressions - Green-Tao (2006).

k equation n ≥ k + 2 variables - Conditional MultidimentionalDirichlet. Best possible dependence excluding ”twin primecase”.

Tamar Ziegler Patterns in primes

Page 33: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Remark: In solving linear equations in primes variables, the nameof the game is

# equations versus # variables

The more variables the easier the question is.

1 equation 2 variables, e.g x1 − x2 = 2. Twin primes.

1 equation 3 variables, e.g the Ternary Goldbach Theorem:every odd number > 5 is a sum of 3 primes: proved byVinogradov (1937) for large enough numbers, very (very)recently proved by Helfgott (13 May 2013) ! or x1 + x3 = 2x2(3 term progressions) proved by Van der Corput (1939). Bothproved using the Hardy Littlewood Circle Method.

Some special cases of k equations f (k) variables - Balog(1992) using the HL circle method.

2 equation 4 variables, e.g x1 + x3 = 2x2; x2 + x4 = 2x3. Thisis 4 term progressions - Green-Tao (2006).

k equation n ≥ k + 2 variables - Conditional MultidimentionalDirichlet. Best possible dependence excluding ”twin primecase”.

Tamar Ziegler Patterns in primes

Page 34: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Remark: In solving linear equations in primes variables, the nameof the game is

# equations versus # variables

The more variables the easier the question is.

1 equation 2 variables, e.g x1 − x2 = 2. Twin primes.

1 equation 3 variables, e.g the Ternary Goldbach Theorem:every odd number > 5 is a sum of 3 primes: proved byVinogradov (1937) for large enough numbers, very (very)recently proved by Helfgott (13 May 2013) ! or x1 + x3 = 2x2(3 term progressions) proved by Van der Corput (1939). Bothproved using the Hardy Littlewood Circle Method.

Some special cases of k equations f (k) variables - Balog(1992) using the HL circle method.

2 equation 4 variables, e.g x1 + x3 = 2x2; x2 + x4 = 2x3. Thisis 4 term progressions - Green-Tao (2006).

k equation n ≥ k + 2 variables - Conditional MultidimentionalDirichlet.

Best possible dependence excluding ”twin primecase”.

Tamar Ziegler Patterns in primes

Page 35: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Remark: In solving linear equations in primes variables, the nameof the game is

# equations versus # variables

The more variables the easier the question is.

1 equation 2 variables, e.g x1 − x2 = 2. Twin primes.

1 equation 3 variables, e.g the Ternary Goldbach Theorem:every odd number > 5 is a sum of 3 primes: proved byVinogradov (1937) for large enough numbers, very (very)recently proved by Helfgott (13 May 2013) ! or x1 + x3 = 2x2(3 term progressions) proved by Van der Corput (1939). Bothproved using the Hardy Littlewood Circle Method.

Some special cases of k equations f (k) variables - Balog(1992) using the HL circle method.

2 equation 4 variables, e.g x1 + x3 = 2x2; x2 + x4 = 2x3. Thisis 4 term progressions - Green-Tao (2006).

k equation n ≥ k + 2 variables - Conditional MultidimentionalDirichlet. Best possible dependence excluding ”twin primecase”.

Tamar Ziegler Patterns in primes

Page 36: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Conditional Multidimentional Dirichlet Theorem

Conditioned on MN(s), and GI (s), one can calculate theasymptotic number of prime solutions to any system of k linearequations with integer coefficients in at least k + 2 variables.

Remark: The method used by Green and Tao to show theexistence of arithmetic progressions in primes can not be used toestablish asymptotics or to handle non homogeneous equations,since it relies on Szemeredi’s theorem which is invalid for nonhomogeneous equations, and can’t provide asymptotics.

Tamar Ziegler Patterns in primes

Page 37: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Conditional Multidimentional Dirichlet Theorem

Conditioned on MN(s), and GI (s), one can calculate theasymptotic number of prime solutions to any system of k linearequations with integer coefficients in at least k + 2 variables.

Remark: The method used by Green and Tao to show theexistence of arithmetic progressions in primes can not be used toestablish asymptotics or to handle non homogeneous equations,since it relies on Szemeredi’s theorem which is invalid for nonhomogeneous equations, and can’t provide asymptotics.

Tamar Ziegler Patterns in primes

Page 38: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

MN(s)

What is MN(s)?

The Mobius Nilsequences conjecture.

Consider the Mobius function

µ(n) =

{(−1)k if n = p1 · · · pk , where pi are distinct primes;0 otherwise.

The Mobius function is related to the normalized prime countingfunction Λ(n) via an identity arising from the mobius inversionformula.

∣∣∣∣∣N∑

n=1

µ(n)

∣∣∣∣∣�A N(logN)−A

Tamar Ziegler Patterns in primes

Page 39: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

MN(s)

What is MN(s)? The Mobius Nilsequences conjecture.

Consider the Mobius function

µ(n) =

{(−1)k if n = p1 · · · pk , where pi are distinct primes;0 otherwise.

The Mobius function is related to the normalized prime countingfunction Λ(n) via an identity arising from the mobius inversionformula.

∣∣∣∣∣N∑

n=1

µ(n)

∣∣∣∣∣�A N(logN)−A

Tamar Ziegler Patterns in primes

Page 40: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

MN(s)

What is MN(s)? The Mobius Nilsequences conjecture.

Consider the Mobius function

µ(n) =

{(−1)k if n = p1 · · · pk , where pi are distinct primes;0 otherwise.

The Mobius function is related to the normalized prime countingfunction Λ(n) via an identity arising from the mobius inversionformula.

∣∣∣∣∣N∑

n=1

µ(n)

∣∣∣∣∣�A N(logN)−A

Tamar Ziegler Patterns in primes

Page 41: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

MN(s)

What is MN(s)? The Mobius Nilsequences conjecture.

Consider the Mobius function

µ(n) =

{(−1)k if n = p1 · · · pk , where pi are distinct primes;0 otherwise.

The Mobius function is related to the normalized prime countingfunction Λ(n) via an identity arising from the mobius inversionformula.

∣∣∣∣∣N∑

n=1

µ(n)

∣∣∣∣∣�A N(logN)−A

Tamar Ziegler Patterns in primes

Page 42: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

MN(s)

Theorem (Davenport (1930s))

For any α ∈ [0, 1]∣∣∣∣∣N∑

n=1

µ(n)e2πinα

∣∣∣∣∣�A N(logN)−A

By the same method: for any polynomial P∣∣∣∣∣N∑

n=1

µ(n)e2πiP(n)

∣∣∣∣∣�A N(logN)−A

Tamar Ziegler Patterns in primes

Page 43: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

MN(s)

Theorem (Davenport (1930s))

For any α ∈ [0, 1]∣∣∣∣∣N∑

n=1

µ(n)e2πinα

∣∣∣∣∣�A N(logN)−A

By the same method: for any polynomial P∣∣∣∣∣N∑

n=1

µ(n)e2πiP(n)

∣∣∣∣∣�A N(logN)−A

Tamar Ziegler Patterns in primes

Page 44: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

We will see soon, that for our purposes, we need similar estimatesfor bracket polynomials. Examples:

n2α, nα{nβ}, nα{n2β}, nα{{nβ}nγ}

Mobius Nilsequence Conjecture (MN(s))

For any bracket polynomial P.∣∣∣∣∣N∑

n=1

µ(n)e2πiP(n)

∣∣∣∣∣�A N(logN)−A

µ does not correlate with bracket polynomial phase functions !

Who is s ? How is this related to nilsequences (what are they)?

Theorem (Green-Tao (2007))

Mobius Nilsequence Conjecture is true.

Tamar Ziegler Patterns in primes

Page 45: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

We will see soon, that for our purposes, we need similar estimatesfor bracket polynomials. Examples:

n2α, nα{nβ}, nα{n2β}, nα{{nβ}nγ}

Mobius Nilsequence Conjecture (MN(s))

For any bracket polynomial P.∣∣∣∣∣N∑

n=1

µ(n)e2πiP(n)

∣∣∣∣∣�A N(logN)−A

µ does not correlate with bracket polynomial phase functions !

Who is s ? How is this related to nilsequences (what are they)?

Theorem (Green-Tao (2007))

Mobius Nilsequence Conjecture is true.

Tamar Ziegler Patterns in primes

Page 46: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

We will see soon, that for our purposes, we need similar estimatesfor bracket polynomials. Examples:

n2α, nα{nβ}, nα{n2β}, nα{{nβ}nγ}

Mobius Nilsequence Conjecture (MN(s))

For any bracket polynomial P.∣∣∣∣∣N∑

n=1

µ(n)e2πiP(n)

∣∣∣∣∣�A N(logN)−A

µ does not correlate with bracket polynomial phase functions !

Who is s ? How is this related to nilsequences (what are they)?

Theorem (Green-Tao (2007))

Mobius Nilsequence Conjecture is true.

Tamar Ziegler Patterns in primes

Page 47: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

We will see soon, that for our purposes, we need similar estimatesfor bracket polynomials. Examples:

n2α, nα{nβ}, nα{n2β}, nα{{nβ}nγ}

Mobius Nilsequence Conjecture (MN(s))

For any bracket polynomial P.∣∣∣∣∣N∑

n=1

µ(n)e2πiP(n)

∣∣∣∣∣�A N(logN)−A

µ does not correlate with bracket polynomial phase functions !

Who is s ? How is this related to nilsequences (what are they)?

Theorem (Green-Tao (2007))

Mobius Nilsequence Conjecture is true.

Tamar Ziegler Patterns in primes

Page 48: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

GI (s)

What is GI (s) ? The Inverse Conjecture for the Us+1[N] Gowersnorm.

What are Gowers norms? Let ZN = Z/NZ, and let f : ZN → D.

Discrete differentiation

Let h ∈ ZN , define the derivative in direction h to be

∆hf (n) := f (n + h)f (n)

Examples:

∆hf (n) ≡ 1 for all h ∈ ZN if and only if f (n) ≡ C .

∆h2∆h1f (n) ≡ 1 for all h1, h2 ∈ ZN if and only if f is a linearphase polynomial, i.e. f (n) = e2πiP(n), where P is a linearpolynomial.

Tamar Ziegler Patterns in primes

Page 49: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

GI (s)

What is GI (s) ? The Inverse Conjecture for the Us+1[N] Gowersnorm.

What are Gowers norms?

Let ZN = Z/NZ, and let f : ZN → D.

Discrete differentiation

Let h ∈ ZN , define the derivative in direction h to be

∆hf (n) := f (n + h)f (n)

Examples:

∆hf (n) ≡ 1 for all h ∈ ZN if and only if f (n) ≡ C .

∆h2∆h1f (n) ≡ 1 for all h1, h2 ∈ ZN if and only if f is a linearphase polynomial, i.e. f (n) = e2πiP(n), where P is a linearpolynomial.

Tamar Ziegler Patterns in primes

Page 50: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

GI (s)

What is GI (s) ? The Inverse Conjecture for the Us+1[N] Gowersnorm.

What are Gowers norms? Let ZN = Z/NZ, and let f : ZN → D.

Discrete differentiation

Let h ∈ ZN , define the derivative in direction h to be

∆hf (n) := f (n + h)f (n)

Examples:

∆hf (n) ≡ 1 for all h ∈ ZN if and only if f (n) ≡ C .

∆h2∆h1f (n) ≡ 1 for all h1, h2 ∈ ZN if and only if f is a linearphase polynomial, i.e. f (n) = e2πiP(n), where P is a linearpolynomial.

Tamar Ziegler Patterns in primes

Page 51: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

GI (s)

What is GI (s) ? The Inverse Conjecture for the Us+1[N] Gowersnorm.

What are Gowers norms? Let ZN = Z/NZ, and let f : ZN → D.

Discrete differentiation

Let h ∈ ZN , define the derivative in direction h to be

∆hf (n) := f (n + h)f (n)

Examples:

∆hf (n) ≡ 1 for all h ∈ ZN if and only if f (n) ≡ C .

∆h2∆h1f (n) ≡ 1 for all h1, h2 ∈ ZN if and only if f is a linearphase polynomial, i.e. f (n) = e2πiP(n), where P is a linearpolynomial.

Tamar Ziegler Patterns in primes

Page 52: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

GI (s)

What is GI (s) ? The Inverse Conjecture for the Us+1[N] Gowersnorm.

What are Gowers norms? Let ZN = Z/NZ, and let f : ZN → D.

Discrete differentiation

Let h ∈ ZN , define the derivative in direction h to be

∆hf (n) := f (n + h)f (n)

Examples:

∆hf (n) ≡ 1 for all h ∈ ZN if and only if f (n) ≡ C .

∆h2∆h1f (n) ≡ 1 for all h1, h2 ∈ ZN if and only if f is a linearphase polynomial, i.e. f (n) = e2πiP(n), where P is a linearpolynomial.

Tamar Ziegler Patterns in primes

Page 53: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

GI (s)

Define the Gowers uniformity norms as follows: Let f : ZN → D.

Gowers norms

‖f ‖2sUs [N] =1

Ns+1

∑n,h1,...,hs∈ZN

∆hs · · ·∆h1f (n)

For s > 1 this is a norm.

Tamar Ziegler Patterns in primes

Page 54: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

‖f ‖2sUs [N] =1

Ns+1

∑n,h1,...,hs∈ZN

∆hs · · ·∆h1f (n)

Remarks:

If ‖f ‖Us [N] = 1 then ∆hs · · ·∆h1f (n) ≡ 1 for allh1, . . . , hk ∈ ZN thus f is a phase polynomial of degree < s.

Conversely, if f is a phase polynomial of degree < s, then‖f ‖Us [N] = 1.

If f correlates with a polynomial phase function of degree < sthen ‖f ‖Us [N] �δ 1.

If f is a random function then ‖f ‖Us [N] = o(1).

Tamar Ziegler Patterns in primes

Page 55: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

‖f ‖2sUs [N] =1

Ns+1

∑n,h1,...,hs∈ZN

∆hs · · ·∆h1f (n)

Remarks:

If ‖f ‖Us [N] = 1 then ∆hs · · ·∆h1f (n) ≡ 1 for allh1, . . . , hk ∈ ZN thus f is a phase polynomial of degree < s.

Conversely, if f is a phase polynomial of degree < s, then‖f ‖Us [N] = 1.

If f correlates with a polynomial phase function of degree < sthen ‖f ‖Us [N] �δ 1.

If f is a random function then ‖f ‖Us [N] = o(1).

Tamar Ziegler Patterns in primes

Page 56: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

‖f ‖2sUs [N] =1

Ns+1

∑n,h1,...,hs∈ZN

∆hs · · ·∆h1f (n)

Remarks:

If ‖f ‖Us [N] = 1 then ∆hs · · ·∆h1f (n) ≡ 1 for allh1, . . . , hk ∈ ZN thus f is a phase polynomial of degree < s.

Conversely, if f is a phase polynomial of degree < s, then‖f ‖Us [N] = 1.

If f correlates with a polynomial phase function of degree < sthen ‖f ‖Us [N] �δ 1.

If f is a random function then ‖f ‖Us [N] = o(1).

Tamar Ziegler Patterns in primes

Page 57: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

‖f ‖2sUs [N] =1

Ns+1

∑n,h1,...,hs∈ZN

∆hs · · ·∆h1f (n)

Remarks:

If ‖f ‖Us [N] = 1 then ∆hs · · ·∆h1f (n) ≡ 1 for allh1, . . . , hk ∈ ZN thus f is a phase polynomial of degree < s.

Conversely, if f is a phase polynomial of degree < s, then‖f ‖Us [N] = 1.

If f correlates with a polynomial phase function of degree < sthen ‖f ‖Us [N] �δ 1.

If f is a random function then ‖f ‖Us [N] = o(1).

Tamar Ziegler Patterns in primes

Page 58: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Recall:

If f correlates with a polynomial phase function of degree < sthen ‖f ‖Us [N] �δ 1.

If f is a random function then ‖f ‖Us [N] = o(1).

The inverse question

What can we say about f if ‖f ‖Us [N] �δ 1 ? Does f correlate witha polynomial phase function?

We will get back to this question soon ...

Tamar Ziegler Patterns in primes

Page 59: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Recall:

If f correlates with a polynomial phase function of degree < sthen ‖f ‖Us [N] �δ 1.

If f is a random function then ‖f ‖Us [N] = o(1).

The inverse question

What can we say about f if ‖f ‖Us [N] �δ 1 ? Does f correlate witha polynomial phase function?

We will get back to this question soon ...

Tamar Ziegler Patterns in primes

Page 60: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Recall:

If f correlates with a polynomial phase function of degree < sthen ‖f ‖Us [N] �δ 1.

If f is a random function then ‖f ‖Us [N] = o(1).

The inverse question

What can we say about f if ‖f ‖Us [N] �δ 1 ?

Does f correlate witha polynomial phase function?

We will get back to this question soon ...

Tamar Ziegler Patterns in primes

Page 61: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Recall:

If f correlates with a polynomial phase function of degree < sthen ‖f ‖Us [N] �δ 1.

If f is a random function then ‖f ‖Us [N] = o(1).

The inverse question

What can we say about f if ‖f ‖Us [N] �δ 1 ? Does f correlate witha polynomial phase function?

We will get back to this question soon ...

Tamar Ziegler Patterns in primes

Page 62: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Recall:

If f correlates with a polynomial phase function of degree < sthen ‖f ‖Us [N] �δ 1.

If f is a random function then ‖f ‖Us [N] = o(1).

The inverse question

What can we say about f if ‖f ‖Us [N] �δ 1 ? Does f correlate witha polynomial phase function?

We will get back to this question soon ...

Tamar Ziegler Patterns in primes

Page 63: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Why are Gowers norms important?

Consider the case of 3 term progressions: Let E ⊂ ZN be of sizeδN for some δ > 0. Let’s try to count 3 term progressions in E .Let 1E (x) be the characteristic function of the set E .

Here is a counting expression∑x ,d∈ZN

1E (x)1E (x + d)1E (x + 2d)

An observation of Gowers: If ‖1E − δ‖U2[N] is small then

∑x ,d∈ZN

1E (x)1E (x + d)1E (x + 2d) ∼ δ3N2

This is the number of 3 term progressions we expect to find in arandom subset of ZN of size δN !

Tamar Ziegler Patterns in primes

Page 64: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Why are Gowers norms important?

Consider the case of 3 term progressions: Let E ⊂ ZN be of sizeδN for some δ > 0. Let’s try to count 3 term progressions in E .Let 1E (x) be the characteristic function of the set E .

Here is a counting expression∑x ,d∈ZN

1E (x)1E (x + d)1E (x + 2d)

An observation of Gowers: If ‖1E − δ‖U2[N] is small then

∑x ,d∈ZN

1E (x)1E (x + d)1E (x + 2d) ∼ δ3N2

This is the number of 3 term progressions we expect to find in arandom subset of ZN of size δN !

Tamar Ziegler Patterns in primes

Page 65: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Why are Gowers norms important?

Consider the case of 3 term progressions: Let E ⊂ ZN be of sizeδN for some δ > 0. Let’s try to count 3 term progressions in E .Let 1E (x) be the characteristic function of the set E .

Here is a counting expression∑x ,d∈ZN

1E (x)1E (x + d)1E (x + 2d)

An observation of Gowers: If ‖1E − δ‖U2[N] is small then

∑x ,d∈ZN

1E (x)1E (x + d)1E (x + 2d) ∼ δ3N2

This is the number of 3 term progressions we expect to find in arandom subset of ZN of size δN !

Tamar Ziegler Patterns in primes

Page 66: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Why are Gowers norms important?

Consider the case of 3 term progressions: Let E ⊂ ZN be of sizeδN for some δ > 0. Let’s try to count 3 term progressions in E .Let 1E (x) be the characteristic function of the set E .

Here is a counting expression∑x ,d∈ZN

1E (x)1E (x + d)1E (x + 2d)

An observation of Gowers: If ‖1E − δ‖U2[N] is small then

∑x ,d∈ZN

1E (x)1E (x + d)1E (x + 2d) ∼ δ3N2

This is the number of 3 term progressions we expect to find in arandom subset of ZN of size δN !

Tamar Ziegler Patterns in primes

Page 67: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Why are Gowers norms important?

Consider the case of 3 term progressions: Let E ⊂ ZN be of sizeδN for some δ > 0. Let’s try to count 3 term progressions in E .Let 1E (x) be the characteristic function of the set E .

Here is a counting expression∑x ,d∈ZN

1E (x)1E (x + d)1E (x + 2d)

An observation of Gowers: If ‖1E − δ‖U2[N] is small then

∑x ,d∈ZN

1E (x)1E (x + d)1E (x + 2d) ∼ δ3N2

This is the number of 3 term progressions we expect to find in arandom subset of ZN of size δN !

Tamar Ziegler Patterns in primes

Page 68: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Why are Gowers norms important?

‖1E − δ‖U2[N] small implies that the number of 3 term progressionsis as in a random set.

There is a similar relation between Us [N] and s + 1 termprogressions.

‖1E − δ‖Us [N] small implies that the number of s+1 termprogressions is as in a random set.∑

x ,d∈ZN

1E (x)1E (x + h) . . . 1E (x + sh) ∼ δs+1N2

Tamar Ziegler Patterns in primes

Page 69: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Why are Gowers norms important?

‖1E − δ‖U2[N] small implies that the number of 3 term progressionsis as in a random set.

There is a similar relation between Us [N] and s + 1 termprogressions.

‖1E − δ‖Us [N] small implies that the number of s+1 termprogressions is as in a random set.∑

x ,d∈ZN

1E (x)1E (x + h) . . . 1E (x + sh) ∼ δs+1N2

Tamar Ziegler Patterns in primes

Page 70: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Why are Gowers norms important?

‖1E − δ‖U2[N] small implies that the number of 3 term progressionsis as in a random set.

There is a similar relation between Us [N] and s + 1 termprogressions.

‖1E − δ‖Us [N] small implies that the number of s+1 termprogressions is as in a random set.

∑x ,d∈ZN

1E (x)1E (x + h) . . . 1E (x + sh) ∼ δs+1N2

Tamar Ziegler Patterns in primes

Page 71: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Why are Gowers norms important?

‖1E − δ‖U2[N] small implies that the number of 3 term progressionsis as in a random set.

There is a similar relation between Us [N] and s + 1 termprogressions.

‖1E − δ‖Us [N] small implies that the number of s+1 termprogressions is as in a random set.∑

x ,d∈ZN

1E (x)1E (x + h) . . . 1E (x + sh) ∼ δs+1N2

Tamar Ziegler Patterns in primes

Page 72: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Why are Gowers norms important?

Even more generally: Let {ψi (~x)}ki=1 be k affine linear integerforms in n variables Suppose no 2 forms are affinely dependent.

Then there is some integer s such that if ‖1E − δ‖Us [N] small then

∑~x∈Zn

N

1E (ψ1(~x))1E (ψ2(~x)) . . . 1E (ψk(~x)) ∼ δkNn

So it is REALLY important to find a good way to test whether‖f ‖Us is small.

Tamar Ziegler Patterns in primes

Page 73: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Why are Gowers norms important?

Even more generally: Let {ψi (~x)}ki=1 be k affine linear integerforms in n variables Suppose no 2 forms are affinely dependent.

Then there is some integer s such that if ‖1E − δ‖Us [N] small then

∑~x∈Zn

N

1E (ψ1(~x))1E (ψ2(~x)) . . . 1E (ψk(~x)) ∼ δkNn

So it is REALLY important to find a good way to test whether‖f ‖Us is small.

Tamar Ziegler Patterns in primes

Page 74: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Why are Gowers norms important?

Even more generally: Let {ψi (~x)}ki=1 be k affine linear integerforms in n variables Suppose no 2 forms are affinely dependent.

Then there is some integer s such that if ‖1E − δ‖Us [N] small then

∑~x∈Zn

N

1E (ψ1(~x))1E (ψ2(~x)) . . . 1E (ψk(~x)) ∼ δkNn

So it is REALLY important to find a good way to test whether‖f ‖Us is small.

Tamar Ziegler Patterns in primes

Page 75: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

The inverse question

Well ... this brings us back to the inverse question:

The inverse question (GI (s))

What can we say about f , if ‖f ‖Us+1[N] �δ 1 ?

Suppose ‖f ‖Us+1[N] �δ 1

s = 0: f correlates with a constant function, trivial.

s = 1: f correlates with a character (a linear phase function),by discrete Fourier analysis.

You might be tempted to think inductively, that If ‖f ‖Us+1 �δ 1then f correlates with a degree ≤ s phase polynomial (recall thatthe converse is true). This turns out to be false ...

Tamar Ziegler Patterns in primes

Page 76: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

The inverse question

Well ... this brings us back to the inverse question:

The inverse question (GI (s))

What can we say about f , if ‖f ‖Us+1[N] �δ 1 ?

Suppose ‖f ‖Us+1[N] �δ 1

s = 0: f correlates with a constant function, trivial.

s = 1: f correlates with a character (a linear phase function),by discrete Fourier analysis.

You might be tempted to think inductively, that If ‖f ‖Us+1 �δ 1then f correlates with a degree ≤ s phase polynomial (recall thatthe converse is true). This turns out to be false ...

Tamar Ziegler Patterns in primes

Page 77: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

The inverse question

Well ... this brings us back to the inverse question:

The inverse question (GI (s))

What can we say about f , if ‖f ‖Us+1[N] �δ 1 ?

Suppose ‖f ‖Us+1[N] �δ 1

s = 0: f correlates with a constant function, trivial.

s = 1: f correlates with a character (a linear phase function),by discrete Fourier analysis.

You might be tempted to think inductively, that If ‖f ‖Us+1 �δ 1then f correlates with a degree ≤ s phase polynomial (recall thatthe converse is true). This turns out to be false ...

Tamar Ziegler Patterns in primes

Page 78: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

The inverse question

Well ... this brings us back to the inverse question:

The inverse question (GI (s))

What can we say about f , if ‖f ‖Us+1[N] �δ 1 ?

Suppose ‖f ‖Us+1[N] �δ 1

s = 0: f correlates with a constant function, trivial.

s = 1: f correlates with a character (a linear phase function),by discrete Fourier analysis.

You might be tempted to think inductively, that If ‖f ‖Us+1 �δ 1then f correlates with a degree ≤ s phase polynomial (recall thatthe converse is true).

This turns out to be false ...

Tamar Ziegler Patterns in primes

Page 79: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

The inverse question

Well ... this brings us back to the inverse question:

The inverse question (GI (s))

What can we say about f , if ‖f ‖Us+1[N] �δ 1 ?

Suppose ‖f ‖Us+1[N] �δ 1

s = 0: f correlates with a constant function, trivial.

s = 1: f correlates with a character (a linear phase function),by discrete Fourier analysis.

You might be tempted to think inductively, that If ‖f ‖Us+1 �δ 1then f correlates with a degree ≤ s phase polynomial (recall thatthe converse is true). This turns out to be false ...

Tamar Ziegler Patterns in primes

Page 80: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

A counter example (essentially due to Furstenberg and Weiss): Let

G =(

1 R R0 1 R0 0 1

)Γ =

(1 Z Z0 1 Z0 0 1

)

G is a 2-step nilpotent Lie Group: [[G ,G ],G ] = {1}.Γ is a lattice in G , and G/Γ is a nilmanifold.

Let x =(

1 x z0 1 y0 0 1

)Γ ∈ G/Γ. Consider the function

F (x) = e2πi(z−{x}y)

F is well defined on G/Γ.

Tamar Ziegler Patterns in primes

Page 81: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

A counter example (essentially due to Furstenberg and Weiss): Let

G =(

1 R R0 1 R0 0 1

)Γ =

(1 Z Z0 1 Z0 0 1

)

G is a 2-step nilpotent Lie Group: [[G ,G ],G ] = {1}.

Γ is a lattice in G , and G/Γ is a nilmanifold.

Let x =(

1 x z0 1 y0 0 1

)Γ ∈ G/Γ. Consider the function

F (x) = e2πi(z−{x}y)

F is well defined on G/Γ.

Tamar Ziegler Patterns in primes

Page 82: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

A counter example (essentially due to Furstenberg and Weiss): Let

G =(

1 R R0 1 R0 0 1

)Γ =

(1 Z Z0 1 Z0 0 1

)

G is a 2-step nilpotent Lie Group: [[G ,G ],G ] = {1}.Γ is a lattice in G , and G/Γ is a nilmanifold.

Let x =(

1 x z0 1 y0 0 1

)Γ ∈ G/Γ. Consider the function

F (x) = e2πi(z−{x}y)

F is well defined on G/Γ.

Tamar Ziegler Patterns in primes

Page 83: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

A counter example (essentially due to Furstenberg and Weiss): Let

G =(

1 R R0 1 R0 0 1

)Γ =

(1 Z Z0 1 Z0 0 1

)

G is a 2-step nilpotent Lie Group: [[G ,G ],G ] = {1}.Γ is a lattice in G , and G/Γ is a nilmanifold.

Let x =(

1 x z0 1 y0 0 1

)Γ ∈ G/Γ. Consider the function

F (x) = e2πi(z−{x}y)

F is well defined on G/Γ.

Tamar Ziegler Patterns in primes

Page 84: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Let a =(

1 α 00 1 β0 0 1

)∈ G .

Consider the function g : Z→ D

g(n) = F (anΓ) = e2πi({nα}nβ+(n2)αβ)

Then‖g‖U3[N] � 1

But if α, β are irrational then, g does not correlate with anyquadratic phase function e2πin

2γ !

Tamar Ziegler Patterns in primes

Page 85: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Let a =(

1 α 00 1 β0 0 1

)∈ G . Consider the function g : Z→ D

g(n) = F (anΓ) = e2πi({nα}nβ+(n2)αβ)

Then‖g‖U3[N] � 1

But if α, β are irrational then, g does not correlate with anyquadratic phase function e2πin

2γ !

Tamar Ziegler Patterns in primes

Page 86: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Let a =(

1 α 00 1 β0 0 1

)∈ G . Consider the function g : Z→ D

g(n) = F (anΓ) = e2πi({nα}nβ+(n2)αβ)

Then‖g‖U3[N] � 1

But if α, β are irrational then, g does not correlate with anyquadratic phase function e2πin

2γ !

Tamar Ziegler Patterns in primes

Page 87: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Let a =(

1 α 00 1 β0 0 1

)∈ G . Consider the function g : Z→ D

g(n) = F (anΓ) = e2πi({nα}nβ+(n2)αβ)

Then‖g‖U3[N] � 1

But if α, β are irrational then, g does not correlate with anyquadratic phase function e2πin

2γ !

Tamar Ziegler Patterns in primes

Page 88: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Inverse U s+1[N]

g(n) = F (anΓ) is an example of a 2-step nilsequence.

In general, if G is an s-step nilpotent Lie group , Γ a lattice, F a”nice function”, and a ∈ G then g(n) = F (anΓ) is an s-stepnilsequence.

The inverse conjecture for the Gowers norm asserts thatnilsequences are the only obstructions to uniformity:

The Inverse Conjecture for the Gowers Norms GI (s)

Let f : ZN → D. Then ‖f ‖Us+1 � 1 if and only if f correlates witha bounded complexity s-step nilsequence.

Polynomial phase functions of degree ≤ s are only a small subsetof the set of degree ≤ s nilsequences !

Tamar Ziegler Patterns in primes

Page 89: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Inverse U s+1[N]

g(n) = F (anΓ) is an example of a 2-step nilsequence.

In general, if G is an s-step nilpotent Lie group , Γ a lattice, F a”nice function”, and a ∈ G then g(n) = F (anΓ) is an s-stepnilsequence.

The inverse conjecture for the Gowers norm asserts thatnilsequences are the only obstructions to uniformity:

The Inverse Conjecture for the Gowers Norms GI (s)

Let f : ZN → D. Then ‖f ‖Us+1 � 1 if and only if f correlates witha bounded complexity s-step nilsequence.

Polynomial phase functions of degree ≤ s are only a small subsetof the set of degree ≤ s nilsequences !

Tamar Ziegler Patterns in primes

Page 90: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Inverse U s+1[N]

g(n) = F (anΓ) is an example of a 2-step nilsequence.

In general, if G is an s-step nilpotent Lie group , Γ a lattice, F a”nice function”, and a ∈ G then g(n) = F (anΓ) is an s-stepnilsequence.

The inverse conjecture for the Gowers norm asserts thatnilsequences are the only obstructions to uniformity:

The Inverse Conjecture for the Gowers Norms GI (s)

Let f : ZN → D. Then ‖f ‖Us+1 � 1 if and only if f correlates witha bounded complexity s-step nilsequence.

Polynomial phase functions of degree ≤ s are only a small subsetof the set of degree ≤ s nilsequences !

Tamar Ziegler Patterns in primes

Page 91: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Inverse U s+1[N]

g(n) = F (anΓ) is an example of a 2-step nilsequence.

In general, if G is an s-step nilpotent Lie group , Γ a lattice, F a”nice function”, and a ∈ G then g(n) = F (anΓ) is an s-stepnilsequence.

The inverse conjecture for the Gowers norm asserts thatnilsequences are the only obstructions to uniformity:

The Inverse Conjecture for the Gowers Norms GI (s)

Let f : ZN → D. Then ‖f ‖Us+1 � 1 if and only if f correlates witha bounded complexity s-step nilsequence.

Polynomial phase functions of degree ≤ s are only a small subsetof the set of degree ≤ s nilsequences !

Tamar Ziegler Patterns in primes

Page 92: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Inverse U s+1[N]

g(n) = F (anΓ) is an example of a 2-step nilsequence.

In general, if G is an s-step nilpotent Lie group , Γ a lattice, F a”nice function”, and a ∈ G then g(n) = F (anΓ) is an s-stepnilsequence.

The inverse conjecture for the Gowers norm asserts thatnilsequences are the only obstructions to uniformity:

The Inverse Conjecture for the Gowers Norms GI (s)

Let f : ZN → D. Then ‖f ‖Us+1 � 1 if and only if f correlates witha bounded complexity s-step nilsequence.

Polynomial phase functions of degree ≤ s are only a small subsetof the set of degree ≤ s nilsequences !

Tamar Ziegler Patterns in primes

Page 93: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

GI (s)

Supporting evidence:

s = 2; Green-Tao (2005).

If ‖f ‖Us+1[N] is sufficiently close to 1, then the conjecture istrue; Alon-Kaufman-Krivelevich-Litsyn and Ron (2005).

Related ergodic results are true; Host-Kra (2002), Z (2004).

What about the finite field analogue for Us+1[Fnp] ?

s = 2; Green-Tao p > 2, Samorodnitsky p = 2 (2005).

For s > 2, a COUNTER EXAMPLE was given by Green-Tao,and independently by Lovett-Meshulam-Samorodnistky(2007).

Tamar Ziegler Patterns in primes

Page 94: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

GI (s)

Supporting evidence:

s = 2; Green-Tao (2005).

If ‖f ‖Us+1[N] is sufficiently close to 1, then the conjecture istrue; Alon-Kaufman-Krivelevich-Litsyn and Ron (2005).

Related ergodic results are true; Host-Kra (2002), Z (2004).

What about the finite field analogue for Us+1[Fnp] ?

s = 2; Green-Tao p > 2, Samorodnitsky p = 2 (2005).

For s > 2, a COUNTER EXAMPLE was given by Green-Tao,and independently by Lovett-Meshulam-Samorodnistky(2007).

Tamar Ziegler Patterns in primes

Page 95: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

GI (s)

Supporting evidence:

s = 2; Green-Tao (2005).

If ‖f ‖Us+1[N] is sufficiently close to 1, then the conjecture istrue; Alon-Kaufman-Krivelevich-Litsyn and Ron (2005).

Related ergodic results are true; Host-Kra (2002), Z (2004).

What about the finite field analogue for Us+1[Fnp] ?

s = 2; Green-Tao p > 2, Samorodnitsky p = 2 (2005).

For s > 2, a COUNTER EXAMPLE was given by Green-Tao,and independently by Lovett-Meshulam-Samorodnistky(2007).

Tamar Ziegler Patterns in primes

Page 96: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

GI (s)

Supporting evidence:

s = 2; Green-Tao (2005).

If ‖f ‖Us+1[N] is sufficiently close to 1, then the conjecture istrue; Alon-Kaufman-Krivelevich-Litsyn and Ron (2005).

Related ergodic results are true; Host-Kra (2002), Z (2004).

What about the finite field analogue for Us+1[Fnp] ?

s = 2; Green-Tao p > 2, Samorodnitsky p = 2 (2005).

For s > 2, a COUNTER EXAMPLE was given by Green-Tao,and independently by Lovett-Meshulam-Samorodnistky(2007).

Tamar Ziegler Patterns in primes

Page 97: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

GI (s)

Supporting evidence:

s = 2; Green-Tao (2005).

If ‖f ‖Us+1[N] is sufficiently close to 1, then the conjecture istrue; Alon-Kaufman-Krivelevich-Litsyn and Ron (2005).

Related ergodic results are true; Host-Kra (2002), Z (2004).

What about the finite field analogue for Us+1[Fnp] ?

s = 2; Green-Tao p > 2, Samorodnitsky p = 2 (2005).

For s > 2, a COUNTER EXAMPLE was given by Green-Tao,and independently by Lovett-Meshulam-Samorodnistky(2007).

Tamar Ziegler Patterns in primes

Page 98: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

GI (s)

Supporting evidence:

s = 2; Green-Tao (2005).

If ‖f ‖Us+1[N] is sufficiently close to 1, then the conjecture istrue; Alon-Kaufman-Krivelevich-Litsyn and Ron (2005).

Related ergodic results are true; Host-Kra (2002), Z (2004).

What about the finite field analogue for Us+1[Fnp] ?

s = 2; Green-Tao p > 2, Samorodnitsky p = 2 (2005).

For s > 2, a COUNTER EXAMPLE was given by Green-Tao,and independently by Lovett-Meshulam-Samorodnistky(2007).

Tamar Ziegler Patterns in primes

Page 99: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

GI (s)

Theorem (Green-Tao-Z (2010))

The Inverse Conjecture for the Gowers Norms is true.

It follows that the (non degenerate) Multidimentional DirichletTheorem is true unconditionally !

Tamar Ziegler Patterns in primes

Page 100: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Mobius Nilsequence Theorem - equivalent formulation MN(s)

The Mobius function µ does not correlate with boundedcomplexity nilsequences:

1

N

∑n≤N

µ(n)F (anΓ)�A1

(logN)A

Inverse Theorem for the Gowers Norms GI (s)

Let f : ZN → D. Then ‖f ‖Us+1 � 1 if and only if f correlates witha bounded complexity s-step nilsequence.

Corollary: ‖µ‖Us+1[N] = o(1).

e.g .1

N2

∑n,d≤N

µ(n)µ(n + d)µ(n + 2d)µ(n + 3d) = o(1)

Tamar Ziegler Patterns in primes

Page 101: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Mobius Nilsequence Theorem - equivalent formulation MN(s)

The Mobius function µ does not correlate with boundedcomplexity nilsequences:

1

N

∑n≤N

µ(n)F (anΓ)�A1

(logN)A

Inverse Theorem for the Gowers Norms GI (s)

Let f : ZN → D. Then ‖f ‖Us+1 � 1 if and only if f correlates witha bounded complexity s-step nilsequence.

Corollary: ‖µ‖Us+1[N] = o(1).

e.g .1

N2

∑n,d≤N

µ(n)µ(n + d)µ(n + 2d)µ(n + 3d) = o(1)

Tamar Ziegler Patterns in primes

Page 102: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Mobius Nilsequence Theorem - equivalent formulation MN(s)

The Mobius function µ does not correlate with boundedcomplexity nilsequences:

1

N

∑n≤N

µ(n)F (anΓ)�A1

(logN)A

Inverse Theorem for the Gowers Norms GI (s)

Let f : ZN → D. Then ‖f ‖Us+1 � 1 if and only if f correlates witha bounded complexity s-step nilsequence.

Corollary: ‖µ‖Us+1[N] = o(1).

e.g .1

N2

∑n,d≤N

µ(n)µ(n + d)µ(n + 2d)µ(n + 3d) = o(1)

Tamar Ziegler Patterns in primes

Page 103: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Mobius Nilsequence Theorem - equivalent formulation MN(s)

The Mobius function µ does not correlate with boundedcomplexity nilsequences:

1

N

∑n≤N

µ(n)F (anΓ)�A1

(logN)A

Inverse Theorem for the Gowers Norms GI (s)

Let f : ZN → D. Then ‖f ‖Us+1 � 1 if and only if f correlates witha bounded complexity s-step nilsequence.

Corollary: ‖µ‖Us+1[N] = o(1).

e.g .1

N2

∑n,d≤N

µ(n)µ(n + d)µ(n + 2d)µ(n + 3d) = o(1)

Tamar Ziegler Patterns in primes

Page 104: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Sketch of proof of multidimensional Dirichlet:

Consider the von-Mangoldt function

Λ(n) =

{log p if n = pk for some prime p and k > 0;0 otherwise.

Recall that we are counting prime values of {ψi (~x)}ki=1 It would begreat if we could show that ‖Λ− 1‖Us+1[N] = o(1). For then wewould have ∑

~x∈[N]n

Λ(ψ1(~x)) · · ·Λ(ψk(~x)) ∼ Nn,

which is what one would expect if one were counting solutions fora random von-Mangold function.

This is unfortunately FALSE (small primes are problematic).

Tamar Ziegler Patterns in primes

Page 105: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Sketch of proof of multidimensional Dirichlet:

Consider the von-Mangoldt function

Λ(n) =

{log p if n = pk for some prime p and k > 0;0 otherwise.

Recall that we are counting prime values of {ψi (~x)}ki=1

It would begreat if we could show that ‖Λ− 1‖Us+1[N] = o(1). For then wewould have ∑

~x∈[N]n

Λ(ψ1(~x)) · · ·Λ(ψk(~x)) ∼ Nn,

which is what one would expect if one were counting solutions fora random von-Mangold function.

This is unfortunately FALSE (small primes are problematic).

Tamar Ziegler Patterns in primes

Page 106: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Sketch of proof of multidimensional Dirichlet:

Consider the von-Mangoldt function

Λ(n) =

{log p if n = pk for some prime p and k > 0;0 otherwise.

Recall that we are counting prime values of {ψi (~x)}ki=1 It would begreat if we could show that ‖Λ− 1‖Us+1[N] = o(1).

For then wewould have ∑

~x∈[N]n

Λ(ψ1(~x)) · · ·Λ(ψk(~x)) ∼ Nn,

which is what one would expect if one were counting solutions fora random von-Mangold function.

This is unfortunately FALSE (small primes are problematic).

Tamar Ziegler Patterns in primes

Page 107: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Sketch of proof of multidimensional Dirichlet:

Consider the von-Mangoldt function

Λ(n) =

{log p if n = pk for some prime p and k > 0;0 otherwise.

Recall that we are counting prime values of {ψi (~x)}ki=1 It would begreat if we could show that ‖Λ− 1‖Us+1[N] = o(1). For then wewould have ∑

~x∈[N]n

Λ(ψ1(~x)) · · ·Λ(ψk(~x)) ∼ Nn,

which is what one would expect if one were counting solutions fora random von-Mangold function.

This is unfortunately FALSE (small primes are problematic).

Tamar Ziegler Patterns in primes

Page 108: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Sketch of proof of multidimensional Dirichlet:

Consider the von-Mangoldt function

Λ(n) =

{log p if n = pk for some prime p and k > 0;0 otherwise.

Recall that we are counting prime values of {ψi (~x)}ki=1 It would begreat if we could show that ‖Λ− 1‖Us+1[N] = o(1). For then wewould have ∑

~x∈[N]n

Λ(ψ1(~x)) · · ·Λ(ψk(~x)) ∼ Nn,

which is what one would expect if one were counting solutions fora random von-Mangold function.

This is unfortunately FALSE (small primes are problematic).

Tamar Ziegler Patterns in primes

Page 109: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

But ...

there is a chance if one looks at Λ along arithmeticprogressions with slowly increasing gap:

Λb,W (n) =φ(W )

WΛ(Wn + b),

where W =∏

p≤w p and we take w = w(N) to be a sufficientlyslowly increasing function of N. (The normalizations is so that1N

∑n≤N Λb,W (n) ∼ 1).

Strategy: show that if (b,W ) = 1 then for any s

‖Λb,W − 1‖Us+1[N] → 0

We then get: ∑~x∈[N]n

Λb,W (ψ1(~x)) · · ·Λb,W (ψk(~x)) ∼ Nn

Tamar Ziegler Patterns in primes

Page 110: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

But ... there is a chance if one looks at Λ along arithmeticprogressions with slowly increasing gap:

Λb,W (n) =φ(W )

WΛ(Wn + b),

where W =∏

p≤w p and we take w = w(N) to be a sufficientlyslowly increasing function of N. (The normalizations is so that1N

∑n≤N Λb,W (n) ∼ 1).

Strategy: show that if (b,W ) = 1 then for any s

‖Λb,W − 1‖Us+1[N] → 0

We then get: ∑~x∈[N]n

Λb,W (ψ1(~x)) · · ·Λb,W (ψk(~x)) ∼ Nn

Tamar Ziegler Patterns in primes

Page 111: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

But ... there is a chance if one looks at Λ along arithmeticprogressions with slowly increasing gap:

Λb,W (n) =φ(W )

WΛ(Wn + b),

where W =∏

p≤w p and we take w = w(N) to be a sufficientlyslowly increasing function of N. (The normalizations is so that1N

∑n≤N Λb,W (n) ∼ 1).

Strategy: show that if (b,W ) = 1 then for any s

‖Λb,W − 1‖Us+1[N] → 0

We then get: ∑~x∈[N]n

Λb,W (ψ1(~x)) · · ·Λb,W (ψk(~x)) ∼ Nn

Tamar Ziegler Patterns in primes

Page 112: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

But ... there is a chance if one looks at Λ along arithmeticprogressions with slowly increasing gap:

Λb,W (n) =φ(W )

WΛ(Wn + b),

where W =∏

p≤w p and we take w = w(N) to be a sufficientlyslowly increasing function of N. (The normalizations is so that1N

∑n≤N Λb,W (n) ∼ 1).

Strategy: show that if (b,W ) = 1 then for any s

‖Λb,W − 1‖Us+1[N] → 0

We then get: ∑~x∈[N]n

Λb,W (ψ1(~x)) · · ·Λb,W (ψk(~x)) ∼ Nn

Tamar Ziegler Patterns in primes

Page 113: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

By the Inverse Theorem for the Gowers Norms it suffices to showthat

Λb,W − 1

does not correlate with nilsequences, but this follows from theMobius Nilsequence Theorem.

(Not so small) Cheat: Inverse theorem applies to boundedfunction, Λb,W isn’t bounded !

To the rescue comes the Green-Tao transference principle, whichallows us to push the inverse theorem from bounded functions tofunction bounded by a pseudorandom function (this is a wholedifferent story ...).

Tamar Ziegler Patterns in primes

Page 114: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

By the Inverse Theorem for the Gowers Norms it suffices to showthat

Λb,W − 1

does not correlate with nilsequences, but this follows from theMobius Nilsequence Theorem.

(Not so small) Cheat: Inverse theorem applies to boundedfunction, Λb,W isn’t bounded !

To the rescue comes the Green-Tao transference principle, whichallows us to push the inverse theorem from bounded functions tofunction bounded by a pseudorandom function (this is a wholedifferent story ...).

Tamar Ziegler Patterns in primes

Page 115: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

By the Inverse Theorem for the Gowers Norms it suffices to showthat

Λb,W − 1

does not correlate with nilsequences, but this follows from theMobius Nilsequence Theorem.

(Not so small) Cheat: Inverse theorem applies to boundedfunction, Λb,W isn’t bounded !

To the rescue comes the Green-Tao transference principle, whichallows us to push the inverse theorem from bounded functions tofunction bounded by a pseudorandom function (this is a wholedifferent story ...).

Tamar Ziegler Patterns in primes

Page 116: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Don’t miss the next lecture:

Intertwining developments in ergodic theory and arithmeticcombinatorics leading to the multidimensional Dirichlet theorem.

Thank you !

Tamar Ziegler Patterns in primes

Page 117: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Don’t miss the next lecture:

Intertwining developments in

ergodic theory and arithmeticcombinatorics leading to the multidimensional Dirichlet theorem.

Thank you !

Tamar Ziegler Patterns in primes

Page 118: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Don’t miss the next lecture:

Intertwining developments in ergodic theory

and arithmeticcombinatorics leading to the multidimensional Dirichlet theorem.

Thank you !

Tamar Ziegler Patterns in primes

Page 119: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Don’t miss the next lecture:

Intertwining developments in ergodic theory and arithmeticcombinatorics

leading to the multidimensional Dirichlet theorem.

Thank you !

Tamar Ziegler Patterns in primes

Page 120: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Don’t miss the next lecture:

Intertwining developments in ergodic theory and arithmeticcombinatorics leading to the multidimensional Dirichlet theorem.

Thank you !

Tamar Ziegler Patterns in primes

Page 121: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

Don’t miss the next lecture:

Intertwining developments in ergodic theory and arithmeticcombinatorics leading to the multidimensional Dirichlet theorem.

Thank you !

Tamar Ziegler Patterns in primes

Page 122: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

The Inverse Theorem for the Gowers Norms GI (s)

Let f : ZN → D. Then ‖f ‖Us+1[N] � 1 if and only if f correlateswith a bounded complexity s-step nilsequence.

Inductively for many h, ‖∆hf ‖Us [N] � 1 =⇒ for many h,∆hf correlates with an (s − 1)-step nilsequence Fh(anhGh/Γh)of bounded complexity.

Clever CS: For many h1 + h2 = h3 + h4 the orbit ofah1 × ah2 × ah3 × ah4 is not equidtributed in the nilmanifoldGh1/Γh1 × Gh2/Γh2 × Gh3/Γh3 × Gh4/Γh4 .

Orbit closures in nilmanifolds are algebraic. Use to find someform of an algebraic relation between the ah (for many h).

Use additive combinatorics to obtain bracket linearity in theparameter h (for many h)

Use the fact that ∆h∆k f = ∆k∆hf (and quantitativeequidistribution) to obtain some symmetry in n, h

Integrate (construction: guess a solution).

Tamar Ziegler Patterns in primes

Page 123: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

The Inverse Theorem for the Gowers Norms GI (s)

Let f : ZN → D. Then ‖f ‖Us+1[N] � 1 if and only if f correlateswith a bounded complexity s-step nilsequence.

Inductively for many h, ‖∆hf ‖Us [N] � 1 =⇒ for many h,∆hf correlates with an (s − 1)-step nilsequence Fh(anhGh/Γh)of bounded complexity.

Clever CS: For many h1 + h2 = h3 + h4 the orbit ofah1 × ah2 × ah3 × ah4 is not equidtributed in the nilmanifoldGh1/Γh1 × Gh2/Γh2 × Gh3/Γh3 × Gh4/Γh4 .

Orbit closures in nilmanifolds are algebraic. Use to find someform of an algebraic relation between the ah (for many h).

Use additive combinatorics to obtain bracket linearity in theparameter h (for many h)

Use the fact that ∆h∆k f = ∆k∆hf (and quantitativeequidistribution) to obtain some symmetry in n, h

Integrate (construction: guess a solution).

Tamar Ziegler Patterns in primes

Page 124: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

The Inverse Theorem for the Gowers Norms GI (s)

Let f : ZN → D. Then ‖f ‖Us+1[N] � 1 if and only if f correlateswith a bounded complexity s-step nilsequence.

Inductively for many h, ‖∆hf ‖Us [N] � 1 =⇒ for many h,∆hf correlates with an (s − 1)-step nilsequence Fh(anhGh/Γh)of bounded complexity.

Clever CS: For many h1 + h2 = h3 + h4 the orbit ofah1 × ah2 × ah3 × ah4 is not equidtributed in the nilmanifoldGh1/Γh1 × Gh2/Γh2 × Gh3/Γh3 × Gh4/Γh4 .

Orbit closures in nilmanifolds are algebraic. Use to find someform of an algebraic relation between the ah (for many h).

Use additive combinatorics to obtain bracket linearity in theparameter h (for many h)

Use the fact that ∆h∆k f = ∆k∆hf (and quantitativeequidistribution) to obtain some symmetry in n, h

Integrate (construction: guess a solution).

Tamar Ziegler Patterns in primes

Page 125: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

The Inverse Theorem for the Gowers Norms GI (s)

Let f : ZN → D. Then ‖f ‖Us+1[N] � 1 if and only if f correlateswith a bounded complexity s-step nilsequence.

Inductively for many h, ‖∆hf ‖Us [N] � 1 =⇒ for many h,∆hf correlates with an (s − 1)-step nilsequence Fh(anhGh/Γh)of bounded complexity.

Clever CS: For many h1 + h2 = h3 + h4 the orbit ofah1 × ah2 × ah3 × ah4 is not equidtributed in the nilmanifoldGh1/Γh1 × Gh2/Γh2 × Gh3/Γh3 × Gh4/Γh4 .

Orbit closures in nilmanifolds are algebraic. Use to find someform of an algebraic relation between the ah (for many h).

Use additive combinatorics to obtain bracket linearity in theparameter h (for many h)

Use the fact that ∆h∆k f = ∆k∆hf (and quantitativeequidistribution) to obtain some symmetry in n, h

Integrate (construction: guess a solution).

Tamar Ziegler Patterns in primes

Page 126: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

The Inverse Theorem for the Gowers Norms GI (s)

Let f : ZN → D. Then ‖f ‖Us+1[N] � 1 if and only if f correlateswith a bounded complexity s-step nilsequence.

Inductively for many h, ‖∆hf ‖Us [N] � 1 =⇒ for many h,∆hf correlates with an (s − 1)-step nilsequence Fh(anhGh/Γh)of bounded complexity.

Clever CS: For many h1 + h2 = h3 + h4 the orbit ofah1 × ah2 × ah3 × ah4 is not equidtributed in the nilmanifoldGh1/Γh1 × Gh2/Γh2 × Gh3/Γh3 × Gh4/Γh4 .

Orbit closures in nilmanifolds are algebraic. Use to find someform of an algebraic relation between the ah (for many h).

Use additive combinatorics to obtain bracket linearity in theparameter h (for many h)

Use the fact that ∆h∆k f = ∆k∆hf (and quantitativeequidistribution) to obtain some symmetry in n, h

Integrate (construction: guess a solution).

Tamar Ziegler Patterns in primes

Page 127: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

The Inverse Theorem for the Gowers Norms GI (s)

Let f : ZN → D. Then ‖f ‖Us+1[N] � 1 if and only if f correlateswith a bounded complexity s-step nilsequence.

Inductively for many h, ‖∆hf ‖Us [N] � 1 =⇒ for many h,∆hf correlates with an (s − 1)-step nilsequence Fh(anhGh/Γh)of bounded complexity.

Clever CS: For many h1 + h2 = h3 + h4 the orbit ofah1 × ah2 × ah3 × ah4 is not equidtributed in the nilmanifoldGh1/Γh1 × Gh2/Γh2 × Gh3/Γh3 × Gh4/Γh4 .

Orbit closures in nilmanifolds are algebraic. Use to find someform of an algebraic relation between the ah (for many h).

Use additive combinatorics to obtain bracket linearity in theparameter h (for many h)

Use the fact that ∆h∆k f = ∆k∆hf (and quantitativeequidistribution) to obtain some symmetry in n, h

Integrate (construction: guess a solution).

Tamar Ziegler Patterns in primes

Page 128: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

The Inverse Theorem for the Gowers Norms GI (s)

Let f : ZN → D. Then ‖f ‖Us+1[N] � 1 if and only if f correlateswith a bounded complexity s-step nilsequence.

Inductively for many h, ‖∆hf ‖Us [N] � 1 =⇒ for many h,∆hf correlates with an (s − 1)-step nilsequence Fh(anhGh/Γh)of bounded complexity.

Clever CS: For many h1 + h2 = h3 + h4 the orbit ofah1 × ah2 × ah3 × ah4 is not equidtributed in the nilmanifoldGh1/Γh1 × Gh2/Γh2 × Gh3/Γh3 × Gh4/Γh4 .

Orbit closures in nilmanifolds are algebraic. Use to find someform of an algebraic relation between the ah (for many h).

Use additive combinatorics to obtain bracket linearity in theparameter h (for many h)

Use the fact that ∆h∆k f = ∆k∆hf (and quantitativeequidistribution) to obtain some symmetry in n, h

Integrate (construction: guess a solution).

Tamar Ziegler Patterns in primes

Page 129: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

U s in finite field geometry

The finite field analogue Us+1[Fnp] is true in the high

characteristic case p ≥ s; Bergelson-Tao-Z, Tao-Z (2008)

The ”correct” analogue is true in low characteristic Tao-Z(2011).

Tamar Ziegler Patterns in primes

Page 130: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

U s in finite field geometry

The finite field analogue Us+1[Fnp] is true in the high

characteristic case p ≥ s; Bergelson-Tao-Z, Tao-Z (2008)

The ”correct” analogue is true in low characteristic Tao-Z(2011).

Tamar Ziegler Patterns in primes

Page 131: Dynamics and prime solutions to linear equationstamarz/prime-EMS.pdf · Dynamics and prime solutions to linear equations Tamar Ziegler Technion/Hebrew University EMS Lecture Series

U s in finite field geometry

The finite field analogue Us+1[Fnp] is true in the high

characteristic case p ≥ s; Bergelson-Tao-Z, Tao-Z (2008)

The ”correct” analogue is true in low characteristic Tao-Z(2011).

Tamar Ziegler Patterns in primes