essentially non-normal numbers for cantor series expansions
TRANSCRIPT
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)
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.
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.
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.
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.
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.
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.
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.
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?
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?
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?
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).
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).
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).
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).
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).
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).
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.
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.
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.
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.
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.
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.
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.
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.
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.
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)
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)
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)
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)
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.
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.
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.
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.
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.
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 →∞.
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 →∞.
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.
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)
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)
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)
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)
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)
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)
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)