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### Transcript of GROUPS IN COMBINATORIAL NUMBER zwsun/ICCM.pdf GROUPS IN COMBINATORIAL NUMBER THEORY 3 Green-Tao...

• Invited talk given at the 4th ICCM (Hangzhou, Dec. 21, 2007)

GROUPS IN COMBINATORIAL NUMBER THEORY

Zhi-Wei Sun

Department of Mathematics Nanjing University

Nanjing 210093, P. R. China [email protected]

http://math.nju.edu.cn/∼zwsun

Shnirel’man’s Theorem (Shnirel’man, 1933). If 0 ∈ A ⊆ N = {0, 1, 2, . . . }

and

σ(A) := inf n>1

|{a ∈ A : 1 6 a 6 n}| n

> 0,

then there is n ∈ Z+ such that

nA = A+ · · ·+A = N.

It follows that there is a constant k ∈ Z+ such that each integer greater

than one can be written as a sum of at most k primes.

Mann’s Theorem (Mann [Ann. of Math., 1942]). Let A and B be subsets

of N containing 0. Then

σ(A+B) > min{1, σ(A) + σ(B)},

where A+B denotes the sumset {a+ b : a ∈ A, b ∈ B}.

1

• 2 ZHI-WEI SUN

Cauchy-Davenport Theorem. Let p be any prime. If A and B are

nonempty subsets of Zp = Z/pZ, then

|A+B| > min{p, |A|+ |B| − 1},

i.e.,

δ(A+B) > min {1, δ(A) + δ(B)− δ({0})} ,

where δ(S) = |S|/|Zp| for any S ⊆ Zp.

Szemeredi’s Theorem (Conjectured by Erdős and Turán in 1936). If

A ⊆ N has positive upper (asymptotic) density, i.e.,

d̄(A) = lim sup n→∞

|{a ∈ A : 1 6 a 6 n}| n

> 0,

then A contains an AP (arithmetic progression) of any given length.

In a long paper published in 2001, W. T. Gowers employed harmonic

analysis and the following deep theorem of G. Freiman to obtain a quan-

titative proof of Szemeredi’s theorem; this is the main reason why Gowers

won the Fields Medal in 1998.

Freiman’s Theorem (Freiman, 1966). Let A be a finite nonempty subset

of Z with |A+A| 6 c|A|. Then A is contained in an n-dimensional AP

Q = Q(a; q1, . . . , qn; l1, . . . , ln) = {a+ x1q1 + · · ·+ xnqn : 0 6 xi < li}

with |Q| 6 c′|A|, where c′ and n depend only on c.

Szemerédi’s theorem plays an important role in the proof of the follow-

ing celebrated result which was also conjectured by Erdős and Turán in

1936.

• GROUPS IN COMBINATORIAL NUMBER THEORY 3

Green-Tao Theorem. There are arbitrarily long APs of primes.

In 2003 E. S. Croot [Ann. of Math.] used the sieve method to confirm

the following long-standing conjecture.

Erdős-Graham Conjecture proved by E. Croot. If {2, 3, . . . } is par-

titioned into finitely many subsets, then one of the subsets contains finitely

many distinct integers x1, . . . , xm satisfying ∑m

k=1 1/xk = 1.

A Conjecture (Z. W. Sun, 2007). If A ⊆ {2, 3, . . . } has positive upper

(asymptotic) density, then there are finitely many distinct elements a1 <

· · · < am of A with ∑m

k=1 1/ak = 1.

The set {3, 5, 7, . . . } has asymptotic density 1/2, and it is known that

1 3

+ 1 5

+ 1 7

+ 1 9

+ 1 11

+ 1 33

+ 1 35

+ 1 45

+ 1 55

+ 1 77

+ 1

105 = 1.

Erdős-Szemerédi Conjecture. Let A be a finite nonempty set of inte-

gers or reals. Then for any ε > 0 there is a constant cε > 0 such that

|A+A|+ |A ·A| > cε|A|2−ε,

where A ·A = {a1a2 : a1, a2 ∈ A}.

Bourgain-Katz-Tao Theorem. Let p be a prime and let ∅ 6= A ⊆ Zp.

If |A| < p1−δ with δ > 0, then there are c(δ) > 0 and ε(δ) > 0 such that

max{|A+A|, |A ·A|} > c(δ)|A|1+ε(δ).

In 1953 Kneser extended the Cauchy-Davenport theorem to general

abelian groups.

• 4 ZHI-WEI SUN

Kneser’s Theorem. Let G be an additive abelian group. Let A and B

be finite nonempty subsets of G, and let H = H(A + B) be the stabilizer

{g ∈ G : g +A+B = A+B}. If |A+B| 6 |A|+ |B| − 1, then

|A+B| = |A+H|+ |B +H| − |H|.

Corollary. Let G be an additive abelian group. Let p(G) be the least order

of a nonzero element of G, or p(G) = +∞ if G is torsion-free. Then, for

any finite nonempty subsets A and B of G, we have

|A+B| > min{p(G), |A|+ |B| − 1}.

Proof. Suppose that |A+B| < |A|+ |B| − 1. Then H = H(A+B) 6= {0}

by Kneser’s theorem. Therefore |H| > p(G) and hence

|A+B| = |A+H|+ |B +H| − |H| > |A+H| > |H| > p(G). �

In 2005, G. Károlyi proved that the corollary remains valid if G is just

a finite group (which may not be abelian).

Erdős-Heilbronn Conjecture (1964). Let p be a prime, and let A be a

subset of the field Zp. Then |2∧A| > min{p, 2|A| − 3}, where

2∧A = AuA = {a+ b : a, b ∈ A, and a 6= b}.

This conjecture is so difficult that it had been open for thirty years

until it was finally confirmed by Dias da Silva and Y. Hamidoune [Bull.

London. Math. Soc. 1994], with the help of the representation theory of

groups.

• GROUPS IN COMBINATORIAL NUMBER THEORY 5

Dias da Silva–Hamidoune Theorem [Bull. London Math. Soc. 1994].

Let F be a field and n be a positive integer. Then for any finite subset A

of F we have

|n∧A| ≥ min{p(F ), n|A| − n2 + 1},

where n∧A denotes the set of all sums of n distinct elements of A.

Corollary. If p is a prime and A ⊆ Zp with |A| > √

4p− 7, then any

element of Zp can be written as a sum of b|A|/2c distinct elements of A.

The following conjecture was originally motivated by graph theory.

Jaeger’s Conjecture. Let F be a finite field with at least 4 elements,

and let A be an invertible n × n matrix with entries in F . There there

exists a vector ~x ∈ Fn such that both ~x and A~x have no zero component.

In 1989 N. Alon and M. Tarsi [Combinarorica, 9(1989)] confirmed the

conjecture in the case when |F | is not a prime. Moreover their method

later resulted in the following powerful principle.

Combinatorial Nullstellensatz [Alon, Comb. Probab. Comput. 1999].

Let A1, . . . , An be finite subsets of a field F with |Ai| > ki ∈ N for

i = 1, . . . , n. If f(x1, . . . , xn) ∈ F [x1, . . . , xn] has degree k1 + · · · + kn,

and [xk11 · · ·xknn ]f(x1, . . . , xn) (the coefficient of x k1 1 · · ·xknn in f) does not

vanish, then there are a1 ∈ A1, . . . , an ∈ An such that f(a1, . . . , an) 6= 0.

The method using the Combinatorial Nullstellensatz is also called the

polynomial method.

• 6 ZHI-WEI SUN

Alon-Nathanson-Ruzsa Theorem [J. Number Theory 56(1996)]). Let

A1, . . . , An be finite nonempty subsets of a field F with |A1| < · · · < |An|.

Then, for the restricted sumset

A1u· · ·uAn = {a1+· · ·+an : a1 ∈ A, . . . , an ∈ An, and ai 6= aj if i 6= j},

we have

|A1 u · · ·uAn| > { p(F ),

n∑ i=1

(|Ai| − i) + 1 } .

The Dias da Silva–Hamidoune Theorem follows from the ANR Theorem

since for each set A with |A| = k > n we can choose subsets A1, . . . , An

of A with cardinalities k − n+ 1, k − n+ 2, . . . , k respectively.

In 2002 Q. H. Hou and Z. W. Sun generalized the Erdős-Heilbronn

conjecture in another direction.

A Result of Q. H. Hou and Z. W. Sun (Acta Arith., 2002). Let

k,m ∈ N and n ∈ Z+. Let F be a field with p(F ) greater than mn and

(k − 1 −m(n − 1))n. If A1, . . . , An are subsets of F with cardinality k,

and Sij ⊆ F and |Sij | 6 m for all i, j = 1, . . . , n with i 6= j, then for the

difference-restricted sumset

C = {a1 + · · ·+ an : a1 ∈ A1, . . . , an ∈ An, ai − aj 6∈ Sij if i 6= j}

we have |C| > (k − 1−m(n− 1))n+ 1.

A key step in the proof of the Hou-Sun result is to show that if k,m, n ∈

• GROUPS IN COMBINATORIAL NUMBER THEORY 7

N, n > 2 and k > m(n− 1) then

[xk−11 · · ·xk−1n ](x1 + · · ·+ xn)((k−1−m(n−1))n ∏

16i min{p(G), |A|+ |B| − 3}.

One of the needed lemmas is the following famous result.

Feit-Thompson Theorem (1963). Every group of odd order is solvable.

V. F. Lev’s Conjecture. Let G be an abelian group, and let A and B

be finite non-empty subsets of G. Then we have

|AuB| > |A|+ |B| − 2− min c∈A+B

νA,B(c),

where

νA,B(c) = |{(a, b) ∈ A×B: a+ b = c}|.

In particular, |AuB| > |A|+ |B|−1 if some c ∈ G can be uniquely written

as a+ b with a ∈ A and b ∈ B.

In 2006 H. Pan and Z. W. Sun applied the Combinatorial Nullstellensatz

in a new way to make the following progress on Lev’s conjecture.

• 8 ZHI-WEI SUN

A Result of H. Pan and Z. W. Sun (Israel J. Math., 2006). Let G be

an abelian group, and let A,B, S be finite non-empty subsets of G with

C = {a+ b: a ∈ A, b ∈ B, and a− b 6∈ S} 6= ∅.

(i) If G is torsion-free or elementary abelian, then

|C| > |A|+ |B| − |S| −min c∈C

νA,B(c).

(ii) If the torsion subgroup Tor(G) = {g ∈ G : g has a finite order} is

cyclic, then

|C| > |A|+ |B| − 2|S| −min c∈C

νA,B(c).

Recently, Z. W. Sun [Finite Fields Appl.] extended the Cauchy-Davenport

theorem in a new direction, and Pan and Sun generalized the Erdős-

Heilbronn conjec