Essentially non-normal numbers for Cantor series expansions

45
Essentially non-normal numbers for Cantor series expansions Roman Nikiforov National Dragomanov University Kyiv, Ukraine Joint work with Dylan Airey (Princeton U.) and Bill Mance (AMU, Poznan)

Transcript of Essentially non-normal numbers for Cantor series expansions

Page 1: Essentially non-normal numbers for Cantor series expansions

Essentially non-normal numbersfor Cantor series expansions

Roman Nikiforov

National Dragomanov UniversityKyiv, Ukraine

Joint work with Dylan Airey (Princeton U.)and Bill Mance (AMU, Poznan)

Page 2: Essentially non-normal numbers for Cantor series expansions

s-adic expansion of numbers

Let s is a base, s ∈ N≥2.

∀x ∈ [0,1] : x =∞∑

i=1

αi(x)

si , αi ∈ {0,1,2, . . . , s − 1},∀i ∈ N

Let B is a finite block of digits (b1b2 . . . bk ), bi ∈ {0,1, . . . , s − 1}

Let Nsn(x ,B) be a number of times a block B appears in the first n

digits of the s-adic expansion of x .

Definitionx is a normal in base s if for all k and blocks B of length k , one has

limn→∞

Nsn(x ,B)

n=

1sk .

x is simply normal in base s if it holds for k = 1.

Page 3: Essentially non-normal numbers for Cantor series expansions

s-adic expansion of numbersLet s is a base, s ∈ N≥2.

∀x ∈ [0,1] : x =∞∑

i=1

αi(x)

si , αi ∈ {0,1,2, . . . , s − 1},∀i ∈ N

Let B is a finite block of digits (b1b2 . . . bk ), bi ∈ {0,1, . . . , s − 1}

Let Nsn(x ,B) be a number of times a block B appears in the first n

digits of the s-adic expansion of x .

Definitionx is a normal in base s if for all k and blocks B of length k , one has

limn→∞

Nsn(x ,B)

n=

1sk .

x is simply normal in base s if it holds for k = 1.

Page 4: Essentially non-normal numbers for Cantor series expansions

s-adic expansion of numbersLet s is a base, s ∈ N≥2.

∀x ∈ [0,1] : x =∞∑

i=1

αi(x)

si , αi ∈ {0,1,2, . . . , s − 1},∀i ∈ N

Let B is a finite block of digits (b1b2 . . . bk ), bi ∈ {0,1, . . . , s − 1}

Let Nsn(x ,B) be a number of times a block B appears in the first n

digits of the s-adic expansion of x .

Definitionx is a normal in base s if for all k and blocks B of length k , one has

limn→∞

Nsn(x ,B)

n=

1sk .

x is simply normal in base s if it holds for k = 1.

Page 5: Essentially non-normal numbers for Cantor series expansions

s-adic expansion of numbersLet s is a base, s ∈ N≥2.

∀x ∈ [0,1] : x =∞∑

i=1

αi(x)

si , αi ∈ {0,1,2, . . . , s − 1},∀i ∈ N

Let B is a finite block of digits (b1b2 . . . bk ), bi ∈ {0,1, . . . , s − 1}

Let Nsn(x ,B) be a number of times a block B appears in the first n

digits of the s-adic expansion of x .

Definitionx is a normal in base s if for all k and blocks B of length k , one has

limn→∞

Nsn(x ,B)

n=

1sk .

x is simply normal in base s if it holds for k = 1.

Page 6: Essentially non-normal numbers for Cantor series expansions

s-adic expansion of numbersLet s is a base, s ∈ N≥2.

∀x ∈ [0,1] : x =∞∑

i=1

αi(x)

si , αi ∈ {0,1,2, . . . , s − 1},∀i ∈ N

Let B is a finite block of digits (b1b2 . . . bk ), bi ∈ {0,1, . . . , s − 1}

Let Nsn(x ,B) be a number of times a block B appears in the first n

digits of the s-adic expansion of x .

Definitionx is a normal in base s if for all k and blocks B of length k , one has

limn→∞

Nsn(x ,B)

n=

1sk .

x is simply normal in base s if it holds for k = 1.

Page 7: Essentially non-normal numbers for Cantor series expansions

s-adic expansion of numbersLet s is a base, s ∈ N≥2.

∀x ∈ [0,1] : x =∞∑

i=1

αi(x)

si , αi ∈ {0,1,2, . . . , s − 1},∀i ∈ N

Let B is a finite block of digits (b1b2 . . . bk ), bi ∈ {0,1, . . . , s − 1}

Let Nsn(x ,B) be a number of times a block B appears in the first n

digits of the s-adic expansion of x .

Definitionx is a normal in base s if for all k and blocks B of length k , one has

limn→∞

Nsn(x ,B)

n=

1sk .

x is simply normal in base s if it holds for k = 1.

Page 8: Essentially non-normal numbers for Cantor series expansions

s-adic expansion of numbersLet s is a base, s ∈ N≥2.

∀x ∈ [0,1] : x =∞∑

i=1

αi(x)

si , αi ∈ {0,1,2, . . . , s − 1},∀i ∈ N

Let B is a finite block of digits (b1b2 . . . bk ), bi ∈ {0,1, . . . , s − 1}

Let Nsn(x ,B) be a number of times a block B appears in the first n

digits of the s-adic expansion of x .

Definitionx is a normal in base s if for all k and blocks B of length k , one has

limn→∞

Nsn(x ,B)

n=

1sk .

x is simply normal in base s if it holds for k = 1.

Page 9: Essentially non-normal numbers for Cantor series expansions

Let Ns is a set of normal numbers in base s.Borel, 1909

λ(Ns) = 1.

λ([0,1] \ Ns) = 0

dimH and Baire category of [0,1] \ Ns?

Page 10: Essentially non-normal numbers for Cantor series expansions

Let Ns is a set of normal numbers in base s.Borel, 1909

λ(Ns) = 1.

λ([0,1] \ Ns) = 0

dimH and Baire category of [0,1] \ Ns?

Page 11: Essentially non-normal numbers for Cantor series expansions

Let Ns is a set of normal numbers in base s.Borel, 1909

λ(Ns) = 1.

λ([0,1] \ Ns) = 0

dimH and Baire category of [0,1] \ Ns?

Page 12: Essentially non-normal numbers for Cantor series expansions

Non-normal numbers

DefinitionLs is a set of essentially non-normal numbers x if for any individualdigit i ∈ {0,1, . . . , s − 1}

limn→∞

Nsn(x , i)

n< lim

n→∞

Nsn(x , i)

n.

1995 Pratsiovytyi, Torbin and 2002 Barreira, Saussol, Schmeling —dimH(Ls) = 1

2005 Albeverio, Pratsiovytyi, Torbin — Ls is of the second Bairecategory.

2003 Olsen, Winter — dimH of the set of divergence point is 1.

2017 Albeverio, Kondratiev, N., Torbin — dimH of the set of essentiallynon-normal numbers is 1 for Q∞-expansion (generalization of Lürothexpansion).

Page 13: Essentially non-normal numbers for Cantor series expansions

Non-normal numbers

DefinitionLs is a set of essentially non-normal numbers x if for any individualdigit i ∈ {0,1, . . . , s − 1}

limn→∞

Nsn(x , i)

n< lim

n→∞

Nsn(x , i)

n.

1995 Pratsiovytyi, Torbin and 2002 Barreira, Saussol, Schmeling —dimH(Ls) = 1

2005 Albeverio, Pratsiovytyi, Torbin — Ls is of the second Bairecategory.

2003 Olsen, Winter — dimH of the set of divergence point is 1.

2017 Albeverio, Kondratiev, N., Torbin — dimH of the set of essentiallynon-normal numbers is 1 for Q∞-expansion (generalization of Lürothexpansion).

Page 14: Essentially non-normal numbers for Cantor series expansions

Non-normal numbers

DefinitionLs is a set of essentially non-normal numbers x if for any individualdigit i ∈ {0,1, . . . , s − 1}

limn→∞

Nsn(x , i)

n< lim

n→∞

Nsn(x , i)

n.

1995 Pratsiovytyi, Torbin and 2002 Barreira, Saussol, Schmeling —dimH(Ls) = 1

2005 Albeverio, Pratsiovytyi, Torbin — Ls is of the second Bairecategory.

2003 Olsen, Winter — dimH of the set of divergence point is 1.

2017 Albeverio, Kondratiev, N., Torbin — dimH of the set of essentiallynon-normal numbers is 1 for Q∞-expansion (generalization of Lürothexpansion).

Page 15: Essentially non-normal numbers for Cantor series expansions

Non-normal numbers

DefinitionLs is a set of essentially non-normal numbers x if for any individualdigit i ∈ {0,1, . . . , s − 1}

limn→∞

Nsn(x , i)

n< lim

n→∞

Nsn(x , i)

n.

1995 Pratsiovytyi, Torbin and 2002 Barreira, Saussol, Schmeling —dimH(Ls) = 1

2005 Albeverio, Pratsiovytyi, Torbin — Ls is of the second Bairecategory.

2003 Olsen, Winter — dimH of the set of divergence point is 1.

2017 Albeverio, Kondratiev, N., Torbin — dimH of the set of essentiallynon-normal numbers is 1 for Q∞-expansion (generalization of Lürothexpansion).

Page 16: Essentially non-normal numbers for Cantor series expansions

Non-normal numbers

DefinitionLs is a set of essentially non-normal numbers x if for any individualdigit i ∈ {0,1, . . . , s − 1}

limn→∞

Nsn(x , i)

n< lim

n→∞

Nsn(x , i)

n.

1995 Pratsiovytyi, Torbin and 2002 Barreira, Saussol, Schmeling —dimH(Ls) = 1

2005 Albeverio, Pratsiovytyi, Torbin — Ls is of the second Bairecategory.

2003 Olsen, Winter — dimH of the set of divergence point is 1.

2017 Albeverio, Kondratiev, N., Torbin — dimH of the set of essentiallynon-normal numbers is 1 for Q∞-expansion (generalization of Lürothexpansion).

Page 17: Essentially non-normal numbers for Cantor series expansions

Non-normal numbers

DefinitionLs is a set of essentially non-normal numbers x if for any individualdigit i ∈ {0,1, . . . , s − 1}

limn→∞

Nsn(x , i)

n< lim

n→∞

Nsn(x , i)

n.

1995 Pratsiovytyi, Torbin and 2002 Barreira, Saussol, Schmeling —dimH(Ls) = 1

2005 Albeverio, Pratsiovytyi, Torbin — Ls is of the second Bairecategory.

2003 Olsen, Winter — dimH of the set of divergence point is 1.

2017 Albeverio, Kondratiev, N., Torbin — dimH of the set of essentiallynon-normal numbers is 1 for Q∞-expansion (generalization of Lürothexpansion).

Page 18: Essentially non-normal numbers for Cantor series expansions

Cantor series expansionIf Q ∈ NN

≥2, then we say that Q is a basic sequence.Given a basic sequence Q = (qn)∞n=1, the Q-Cantor series expansionof a real number x ∈ [0,1] is the (unique) expansion of the form

x =∞∑

n=1

En

q1q2 · · · qn

where En is in {0,1, . . . ,qn − 1} for n ≥ 1 with En 6= qn − 1 infinitelyoften. We abbreviate it with the notation x = E1E2E3 . . . w.r.t. Q.

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Normal numbers for Cantor series expansionsLet NQ

n (B, x) denote the number of occurrences of the block B in thedigits of the Q-Cantor series expansion of x up to position n.

For a basic sequence Q = (qn), a block B = (b1,b2, · · · ,b`), and anatural number j , define

Ij(B,Q) =

{1 if b1 < qj ,b2 < qj+1, · · · ,b` < qj+`−1

0 otherwise

and let

Qn(B) =n∑

j=1

Ij(B,Q)

qjqj+1 · · · qj+`−1.

A real number x is Q-normal if for all blocks B such thatlim

n→∞Qn(B) =∞

limn→∞

NQn (B, x)

Qn(B)= 1.

Page 20: Essentially non-normal numbers for Cantor series expansions

Normal numbers for Cantor series expansionsLet NQ

n (B, x) denote the number of occurrences of the block B in thedigits of the Q-Cantor series expansion of x up to position n.For a basic sequence Q = (qn), a block B = (b1,b2, · · · ,b`), and anatural number j , define

Ij(B,Q) =

{1 if b1 < qj ,b2 < qj+1, · · · ,b` < qj+`−1

0 otherwise

and let

Qn(B) =n∑

j=1

Ij(B,Q)

qjqj+1 · · · qj+`−1.

A real number x is Q-normal if for all blocks B such thatlim

n→∞Qn(B) =∞

limn→∞

NQn (B, x)

Qn(B)= 1.

Page 21: Essentially non-normal numbers for Cantor series expansions

Normal numbers for Cantor series expansionsLet NQ

n (B, x) denote the number of occurrences of the block B in thedigits of the Q-Cantor series expansion of x up to position n.For a basic sequence Q = (qn), a block B = (b1,b2, · · · ,b`), and anatural number j , define

Ij(B,Q) =

{1 if b1 < qj ,b2 < qj+1, · · · ,b` < qj+`−1

0 otherwise

and let

Qn(B) =n∑

j=1

Ij(B,Q)

qjqj+1 · · · qj+`−1.

A real number x is Q-normal if for all blocks B such thatlim

n→∞Qn(B) =∞

limn→∞

NQn (B, x)

Qn(B)= 1.

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Essentially non-normal numbers for Cantor series expansions

A real number x is Q-essentially non-normal if for all blocks B suchlim

n→∞Qn(B) =∞ the limit

limn→∞

NQn (B, x)

Qn(B)

does not exist.

Let L(Q) is a set of Q-essentially non-normal numbers.

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Essentially non-normal numbers for Cantor series expansions

A real number x is Q-essentially non-normal if for all blocks B suchlim

n→∞Qn(B) =∞ the limit

limn→∞

NQn (B, x)

Qn(B)

does not exist.

Let L(Q) is a set of Q-essentially non-normal numbers.

Page 24: Essentially non-normal numbers for Cantor series expansions

Let X be a subshift of NN≥2, measure µ is fully supported in X , let basic

sequence Q be a generic point for the dynamical system (X ,T , µ).

Theorem

dimH(L(Q)) = 1.

TheoremL(Q) is the set of second Baire category.

Page 25: Essentially non-normal numbers for Cantor series expansions

Let X be a subshift of NN≥2, measure µ is fully supported in X , let basic

sequence Q be a generic point for the dynamical system (X ,T , µ).

Theorem

dimH(L(Q)) = 1.

TheoremL(Q) is the set of second Baire category.

Page 26: Essentially non-normal numbers for Cantor series expansions

Let X be a subshift of NN≥2, measure µ is fully supported in X , let basic

sequence Q be a generic point for the dynamical system (X ,T , µ).

Theorem

dimH(L(Q)) = 1.

TheoremL(Q) is the set of second Baire category.

Page 27: Essentially non-normal numbers for Cantor series expansions

Normal numbers along arithmetic progressionLet m ∈ N and 0 ≤ r ≤ m − 1.

If

x =∞∑

i=1

αi(x)

si

is normal in base s, then

xm,r =∞∑

t=0

αmt+r (x)

st+1

(Furstenberg 1967)

For continued fraction expansion it does not true. (Vandehey2016)

For Cantor series expansion it does not true (Airey, Mance 2017)

Page 28: Essentially non-normal numbers for Cantor series expansions

Normal numbers along arithmetic progressionLet m ∈ N and 0 ≤ r ≤ m − 1.If

x =∞∑

i=1

αi(x)

si

is normal in base s, then

xm,r =∞∑

t=0

αmt+r (x)

st+1

(Furstenberg 1967)

For continued fraction expansion it does not true. (Vandehey2016)

For Cantor series expansion it does not true (Airey, Mance 2017)

Page 29: Essentially non-normal numbers for Cantor series expansions

Normal numbers along arithmetic progressionLet m ∈ N and 0 ≤ r ≤ m − 1.If

x =∞∑

i=1

αi(x)

si

is normal in base s, then

xm,r =∞∑

t=0

αmt+r (x)

st+1

(Furstenberg 1967)

For continued fraction expansion it does not true. (Vandehey2016)

For Cantor series expansion it does not true (Airey, Mance 2017)

Page 30: Essentially non-normal numbers for Cantor series expansions

Normal numbers along arithmetic progressionLet m ∈ N and 0 ≤ r ≤ m − 1.If

x =∞∑

i=1

αi(x)

si

is normal in base s, then

xm,r =∞∑

t=0

αmt+r (x)

st+1

(Furstenberg 1967)

For continued fraction expansion it does not true. (Vandehey2016)

For Cantor series expansion it does not true (Airey, Mance 2017)

Page 31: Essentially non-normal numbers for Cantor series expansions

Essentially non-normal numbers along arithmetic progressionFor m ∈ N and 0 ≤ r ≤ m − 1 we define the basic sequenceΛm,r (Q) := (qmt+r )∞t=0.If x = E1E2 · · · w.r.t. Q, then let

ΥQ,m,r (x) := Er Em+r E2m+r · · · w.r.t. Λm,r (Q).

LetLm,r (Q) = {ΥQ,m,r (x), x ∈ LQ}.

Let the dynamical system (X ,T , µ) be weak-mixing.

TheoremLm,r (Q) has only ΛAm,r (Q)-essentially non-normal numbers.

Theorem

dimH(Lm,r (Q)) = 1.

Page 32: Essentially non-normal numbers for Cantor series expansions

Essentially non-normal numbers along arithmetic progressionFor m ∈ N and 0 ≤ r ≤ m − 1 we define the basic sequenceΛm,r (Q) := (qmt+r )∞t=0.If x = E1E2 · · · w.r.t. Q, then let

ΥQ,m,r (x) := Er Em+r E2m+r · · · w.r.t. Λm,r (Q).

LetLm,r (Q) = {ΥQ,m,r (x), x ∈ LQ}.

Let the dynamical system (X ,T , µ) be weak-mixing.

TheoremLm,r (Q) has only ΛAm,r (Q)-essentially non-normal numbers.

Theorem

dimH(Lm,r (Q)) = 1.

Page 33: Essentially non-normal numbers for Cantor series expansions

Essentially non-normal numbers along arithmetic progressionFor m ∈ N and 0 ≤ r ≤ m − 1 we define the basic sequenceΛm,r (Q) := (qmt+r )∞t=0.If x = E1E2 · · · w.r.t. Q, then let

ΥQ,m,r (x) := Er Em+r E2m+r · · · w.r.t. Λm,r (Q).

LetLm,r (Q) = {ΥQ,m,r (x), x ∈ LQ}.

Let the dynamical system (X ,T , µ) be weak-mixing.

TheoremLm,r (Q) has only ΛAm,r (Q)-essentially non-normal numbers.

Theorem

dimH(Lm,r (Q)) = 1.

Page 34: Essentially non-normal numbers for Cantor series expansions

Essentially non-normal numbers along arithmetic progressionFor m ∈ N and 0 ≤ r ≤ m − 1 we define the basic sequenceΛm,r (Q) := (qmt+r )∞t=0.If x = E1E2 · · · w.r.t. Q, then let

ΥQ,m,r (x) := Er Em+r E2m+r · · · w.r.t. Λm,r (Q).

LetLm,r (Q) = {ΥQ,m,r (x), x ∈ LQ}.

Let the dynamical system (X ,T , µ) be weak-mixing.

TheoremLm,r (Q) has only ΛAm,r (Q)-essentially non-normal numbers.

Theorem

dimH(Lm,r (Q)) = 1.

Page 35: Essentially non-normal numbers for Cantor series expansions

Essentially non-normal numbers along arithmetic progressionM = (mt )t is an increasing sequence of positive integers.Given a sequence M, we define the basic sequenceΛM(Q) := (qmt )

∞t=1.

If x = E1E2 · · · w.r.t. Q, then let

ΥQ,M(x) := Em1Em2Em3 · · · w.r.t. ΛM(Q).

For m ∈ N and 0 ≤ r ≤ m − 1 let Am,r := (mt + r)∞t=0.

LetLm,r (Q) = {ΥQ,Am,r (x), x ∈ LQ}.

Let the dynamical system (X ,T , µ) be weak-mixing.

Theorem

dimH(Lm,r (Q)) = 1.

TheoremSet Lm,r (Q) is of second Baire category.

Page 36: Essentially non-normal numbers for Cantor series expansions

Idea of proofWe construct subset of L(Q)

Ls(Q) ={

x : x ∈ (0,1),

x = α1,1α1,2 . . . α1,4sγ1,1γ1,201︸ ︷︷ ︸first group

α2,1α2,2 . . . α2,8sγ2,1γ2,2γ2,3γ2,40011︸ ︷︷ ︸second group

. . .

αk ,1αk ,2 . . . αk ,2k+1sγk ,1 . . . γk ,2k

2k−1︷ ︸︸ ︷0 . . . 0

2k−1︷ ︸︸ ︷1 . . . 1︸ ︷︷ ︸

k -th group

. . .,

where αk ,j ∈ {0,1, . . . ,qk ,j − 1},∀k ∈ N}.

Let y = γ1,1γ1,2γ2,1γ2,2γ2,3γ2,4 . . . γk ,1 . . . γk ,2k . . . is normal number forthe basic sequence Ps = (Q1,sQ2,s . . .), where Qi,s is a part of basicsequence Q in which positions digits of y are standing.

dimH(Ls(Q))→ 1, s →∞.

Page 37: Essentially non-normal numbers for Cantor series expansions

Idea of proofWe construct subset of L(Q)

Ls(Q) ={

x : x ∈ (0,1),

x = α1,1α1,2 . . . α1,4sγ1,1γ1,201︸ ︷︷ ︸first group

α2,1α2,2 . . . α2,8sγ2,1γ2,2γ2,3γ2,40011︸ ︷︷ ︸second group

. . .

αk ,1αk ,2 . . . αk ,2k+1sγk ,1 . . . γk ,2k

2k−1︷ ︸︸ ︷0 . . . 0

2k−1︷ ︸︸ ︷1 . . . 1︸ ︷︷ ︸

k -th group

. . .,

where αk ,j ∈ {0,1, . . . ,qk ,j − 1},∀k ∈ N}.

Let y = γ1,1γ1,2γ2,1γ2,2γ2,3γ2,4 . . . γk ,1 . . . γk ,2k . . . is normal number forthe basic sequence Ps = (Q1,sQ2,s . . .), where Qi,s is a part of basicsequence Q in which positions digits of y are standing.

dimH(Ls(Q))→ 1, s →∞.

Page 38: Essentially non-normal numbers for Cantor series expansions

For proof dimH(Lm,r (Q)) = 1 we use

Theorem (V. Bergelson, J. Vandehey, paper under preparation)Let (X ,T , µ) is weak mixing.If x ∈ X is a normal number with symbolic expansion [a1,a2,a3, . . .]and y = [am,am+r ,am+2r , ...], then the frequency of any block B in yexists.

Page 39: Essentially non-normal numbers for Cantor series expansions

Examples

s-adic expansionµ is Bernoulli measureBasic sequence Q is shifted by 2 the Thue–Morse sequence

2332322332232332 . . .

Basic sequence Q is a shifted by 1 continued fraction normalnumber. For example, Adler–Keane–Smorodinsky number:concatenation of continued fractions of the rationals

12,13,23,14,24,34, . . .

Q is a shifted by 2 the Copeland-Erdos number

457911333539311 . . .

Q is a shifted by 2 Lüroth normal number or β-normal numbers,β = 1+

√5

2 (examples in Madritsch, Mance 2016)

Page 40: Essentially non-normal numbers for Cantor series expansions

Exampless-adic expansion

µ is Bernoulli measureBasic sequence Q is shifted by 2 the Thue–Morse sequence

2332322332232332 . . .

Basic sequence Q is a shifted by 1 continued fraction normalnumber. For example, Adler–Keane–Smorodinsky number:concatenation of continued fractions of the rationals

12,13,23,14,24,34, . . .

Q is a shifted by 2 the Copeland-Erdos number

457911333539311 . . .

Q is a shifted by 2 Lüroth normal number or β-normal numbers,β = 1+

√5

2 (examples in Madritsch, Mance 2016)

Page 41: Essentially non-normal numbers for Cantor series expansions

Exampless-adic expansionµ is Bernoulli measure

Basic sequence Q is shifted by 2 the Thue–Morse sequence

2332322332232332 . . .

Basic sequence Q is a shifted by 1 continued fraction normalnumber. For example, Adler–Keane–Smorodinsky number:concatenation of continued fractions of the rationals

12,13,23,14,24,34, . . .

Q is a shifted by 2 the Copeland-Erdos number

457911333539311 . . .

Q is a shifted by 2 Lüroth normal number or β-normal numbers,β = 1+

√5

2 (examples in Madritsch, Mance 2016)

Page 42: Essentially non-normal numbers for Cantor series expansions

Exampless-adic expansionµ is Bernoulli measureBasic sequence Q is shifted by 2 the Thue–Morse sequence

2332322332232332 . . .

Basic sequence Q is a shifted by 1 continued fraction normalnumber. For example, Adler–Keane–Smorodinsky number:concatenation of continued fractions of the rationals

12,13,23,14,24,34, . . .

Q is a shifted by 2 the Copeland-Erdos number

457911333539311 . . .

Q is a shifted by 2 Lüroth normal number or β-normal numbers,β = 1+

√5

2 (examples in Madritsch, Mance 2016)

Page 43: Essentially non-normal numbers for Cantor series expansions

Exampless-adic expansionµ is Bernoulli measureBasic sequence Q is shifted by 2 the Thue–Morse sequence

2332322332232332 . . .

Basic sequence Q is a shifted by 1 continued fraction normalnumber. For example, Adler–Keane–Smorodinsky number:concatenation of continued fractions of the rationals

12,13,23,14,24,34, . . .

Q is a shifted by 2 the Copeland-Erdos number

457911333539311 . . .

Q is a shifted by 2 Lüroth normal number or β-normal numbers,β = 1+

√5

2 (examples in Madritsch, Mance 2016)

Page 44: Essentially non-normal numbers for Cantor series expansions

Exampless-adic expansionµ is Bernoulli measureBasic sequence Q is shifted by 2 the Thue–Morse sequence

2332322332232332 . . .

Basic sequence Q is a shifted by 1 continued fraction normalnumber. For example, Adler–Keane–Smorodinsky number:concatenation of continued fractions of the rationals

12,13,23,14,24,34, . . .

Q is a shifted by 2 the Copeland-Erdos number

457911333539311 . . .

Q is a shifted by 2 Lüroth normal number or β-normal numbers,β = 1+

√5

2 (examples in Madritsch, Mance 2016)

Page 45: Essentially non-normal numbers for Cantor series expansions

Exampless-adic expansionµ is Bernoulli measureBasic sequence Q is shifted by 2 the Thue–Morse sequence

2332322332232332 . . .

Basic sequence Q is a shifted by 1 continued fraction normalnumber. For example, Adler–Keane–Smorodinsky number:concatenation of continued fractions of the rationals

12,13,23,14,24,34, . . .

Q is a shifted by 2 the Copeland-Erdos number

457911333539311 . . .

Q is a shifted by 2 Lüroth normal number or β-normal numbers,β = 1+

√5

2 (examples in Madritsch, Mance 2016)