Counting patterns in digital expansions

Transcription

1 Counting patterns in digital expansions Manfred G. Madritsch Department for Analysis and Computational Number Theory Graz University of Technology Séminaire ERNEST Dynamique, Arithmétique, Combinatoire Supported by the Austrian Research Fund (FWF), Project S / 70

2 Outline Motivation Number systems and additive functions Distributional properties Normal numbers Method Transfer of the problems Fourier transform Weyl Sums Division of the expansion Results Arithmetic sum Asymptotic distribution Construction of normal numbers 2 / 70

3 Number systems Let R be an integral domain, b R, and N = {n 1,..., n m } R. Then we call the pair (b, N ) a number system in R if every g R admits a unique and finite representation of the form g = h a j (g)b j with a i (g) N for i = 0,..., h (1) j=0 and a h (g) 0 if h 0. We call b the base and N the set of digits. 3 / 70

4 Examples for number systems b Z, b 2 and N := {0, 1,..., b 1}, then (b, N ) is a number system in Z. B F q [X ] a polynomial, deg B > 1, N := {P F q [X ] : deg P < deg B}. Then (B, N ) is a number system in F q [X ]. Let β be an algebraic integer over Z. Furthermore let b Z[β] and N := {0, 1,..., N(b) 1}. Then under some mild conditions the pair (b, N ) is a number system in Z[β]. 4 / 70

5 Examples for number systems b Z, b 2 and N := {0, 1,..., b 1}, then (b, N ) is a number system in Z. B F q [X ] a polynomial, deg B > 1, N := {P F q [X ] : deg P < deg B}. Then (B, N ) is a number system in F q [X ]. Let β be an algebraic integer over Z. Furthermore let b Z[β] and N := {0, 1,..., N(b) 1}. Then under some mild conditions the pair (b, N ) is a number system in Z[β]. 4 / 70

6 Examples for number systems b Z, b 2 and N := {0, 1,..., b 1}, then (b, N ) is a number system in Z. B F q [X ] a polynomial, deg B > 1, N := {P F q [X ] : deg P < deg B}. Then (B, N ) is a number system in F q [X ]. Let β be an algebraic integer over Z. Furthermore let b Z[β] and N := {0, 1,..., N(b) 1}. Then under some mild conditions the pair (b, N ) is a number system in Z[β]. 4 / 70

7 Additive functions Let R be an integral domain and (b, N ) be a number system in this domain. Then we call a function f : R R b-additive, if for g as in (1) we have that h f (g) = f (a k b k ). k=0 Moreover we call it strictly b-additive, if for g as in (1) we have that h f (g) = f (a k ). k=0 5 / 70

8 Additive functions Let R be an integral domain and (b, N ) be a number system in this domain. Then we call a function f : R R b-additive, if for g as in (1) we have that h f (g) = f (a k b k ). k=0 Moreover we call it strictly b-additive, if for g as in (1) we have that h f (g) = f (a k ). k=0 5 / 70

9 The sum-of-digits function A very simple example of a strictly b-additive function is the sum-of-digits function s b, which is defined by s b (g) = h k=0 a k for g as in (1). 6 / 70

10 Outline Motivation Number systems and additive functions Distributional properties Normal numbers Method Transfer of the problems Fourier transform Weyl Sums Division of the expansion Results Arithmetic sum Asymptotic distribution Construction of normal numbers 7 / 70

11 Delange s Result Theorem Delange (1975) n N s q (n) = q 1 N log 2 q N + NF ( log q N ), where log q is the logarithm to base q and F is a 1-periodic, continuous and nowhere differentiable function. 8 / 70

12 Distribution result for integers Let f be a q-additive function in N such that f (aq k ) = O(1) as k and a N. Furthermore let m k,q := 1 f (aq k ), q a N σk,q 2 := 1 f 2 (aq k ) m 2 q k,q, a N and M q (x) := with N = [log q x].l N m k,q, Dq(x) 2 = k=0 N k=0 σ 2 k,q 9 / 70

13 Distribution result for integers Theorem Bassily and Katai (1995) Assume that D q (x)/(log x) 1/3 as x and let p(x) be a polynomial with integer coefficients, degree d and positive leading term. Then, as x, { 1 x # n < x f (p(n)) M q (x d ) D q (x d ) where Φ is the normal distribution function. } < y Φ(y), 10 / 70

14 Distribution result for Gaussian integers Let f be a b-additive function in Z[i] such that f (ab k ) = O(1) as k and a N. Furthermore let m k,b := 1 N(b) a N f (ab k ), σ 2 k,b := 1 N(b) f 2 (ab k ) mk,b, 2 a N and M b (x) := with L = [log N(b) x]. L m k,q, Db(x) 2 = k=0 L k=0 σ 2 k,q 11 / 70

15 Distribution result for Gaussian integers Theorem Gittenberger and Thuswaldner (2000) Assume that D b (x)/(log x) 1/3 as x and let p Z[i][X ] be a polynomial of degree d. Then, as T, { 1 # {z N(z) N} # N(z) N f (p(z)) M b (N d ) D b (N d ) where Φ is the normal distribution function. } < y Φ(y), 12 / 70

16 Outline Motivation Number systems and additive functions Distributional properties Normal numbers Method Transfer of the problems Fourier transform Weyl Sums Division of the expansion Results Arithmetic sum Asymptotic distribution Construction of normal numbers 13 / 70

17 Continuation We extend our number system onto K the completion of the field of quotients K of R. Then we get that every γ K has a (not necessarily unique) representation of the shape γ = l(γ) j= a j (γ)b j (a j (γ) N ). 14 / 70

18 Fundamental domain In this context the fundamental domain F indicates the properties of this extension. It is defined as all numbers with zero in the integer part of their b-ary representation, i.e., { } F := γ K γ = j 1 a j b j, a j N. 15 / 70

19 Block count Let θ K be such that θ = j 1 a j b j F. Then for d 1... d r N r being a block of digits of length r we denote by N (θ; d 1... d r ; N) the number of occurrences of this block in the first N digits of θ. Thus N (θ; d 1... d r ; N) := #{0 n < N : d 1 = a n+1,..., d r = a n+r }. 16 / 70

20 Normal number Now we call θ normal in (b, N ) if for every r 1 we have that R N (θ) = R N,r (θ) := sup 1 d 1...d r N N (θ; d 1... d r ; N) 1 N r = o(1) where the supremum is taken over all possible blocks d 1... d r N r of length r. 17 / 70

21 Construction of normal numbers In order to construct a normal number in base b we often take a strictly increasing sequence (s n ) n 1 of real numbers and concatenate its values written in base b. Thus we define θ((s n ) n 1 ) := 0. s 1 s 2 s 3 s 4 s / 70

22 Constructions of normal numbers Theorem Champernowne (1933) is normal to base 10. θ((n) n 1 ) = Theorem Copeland and Erdős (1946) Let s n N. If δ > 0 N N : #{s n : s n N} N δ, then θ((s n ) n 1 ) is normal. 19 / 70

23 Outline Motivation Number systems and additive functions Distributional properties Normal numbers Method Transfer of the problems Fourier transform Weyl Sums Division of the expansion Results Arithmetic sum Asymptotic distribution Construction of normal numbers 20 / 70

24 Our goals Let p be an arbitrary function from the reals into the reals and f be an q-additive function. Then we want to estimate the following f (p(n)) n N { # n N : f (p(n)) µ f σ f } < y N (θ (p(n)) n 1 ), d 1... d r, N) 21 / 70

25 Indicator function where I l,a (x) = 1 ( x q + ψ q a + 1 ) l+1 q { 1 if a l (x) = a, = 0 else, ψ(x) = x x 1 2 = {x} 1 2. ( x ψ q a ) l+1 q 22 / 70

26 Rewriting our goals For the arithmetic sum we get, that f (p(n)) = n N n N q 1 = a=0 log q p(n) l=0 log q p(n) l=0 q 1 f (aq l )I l,a (p(n)) a=0 f (aq l )I l,a (p(n)). n N 23 / 70

27 Transferation of the problem For the asymptotic distribution result there is more work to be done. 1. Truncate the range of the function f. 2. Consideration of the moments. 3. Getting rid of the polynomial. 24 / 70

28 Transferation: Truncation Let 0 A B L be well chosen. Then define the truncated function f to be f (p(n)) = B f (a j (p(n))b j ). j=a We easily get that max n N f (p(n)) M(T d ) D(T d ) f (p(n)) M (T d ) D (T d ) / 70

29 Transferation: Moments Thus it suffices to consider { 1 N # n N f (p(n)) M (T d ) D (T d ) } < y Φ(y). and by the Frechet-Shohat Theorem this holds if and only if the moments ξ k (T ) := 1 ( ) f (p(n)) M (T d k ) N D (T d ) n N converge to the moments of the normal law. 26 / 70

30 Transferation: Polynomial Finally we want to get rid of the polynomial p and this is done by instead of considering the moments ξ k (T ) := 1 N ( f (p(n)) M (T d ) D (T d ) n N ) k considering the following moments η k (T ) := 1 N ( ) f (n) M (T d k ). D (T d ) n N 27 / 70

31 Transferation: The central tool Now we are left with showing that ξ k (T ) η k (T ) 0 for T. In particular we are left with estimating # { n N alj (p(n)) = a j, j = 1,..., h } = n N where A l 1 < l 2 < < l h B. h I lj,a j (p(n)), j=1 28 / 70

32 Block counting For proving that one of the constructions above really yields a normal number one counts the number of occurrences of a pattern within the expansion and ignores the number occurring between two expansions. Thus we need to estimate n N log q p(n) r+1 l=0 r 1 I l+j,aj (p(n)). j=0 29 / 70

33 Block counting For proving that one of the constructions above really yields a normal number one counts the number of occurrences of a pattern within the expansion and ignores the number occurring between two expansions. Thus we need to estimate n N log q p(n) r+1 l=0 r 1 I l+j,aj (p(n)). j=0 29 / 70

34 Counting the patterns Thus we have to count the following patterns : q 1 a=0 n N log q p(n) l=0 f (aq l )I l,a (p(n)), n N n N j=1 log q p(n) r+1 l=0 h I lj,a j (p(n)), r 1 I l+j,aj (p(n)). j=0 30 / 70

35 Outline Motivation Number systems and additive functions Distributional properties Normal numbers Method Transfer of the problems Fourier transform Weyl Sums Division of the expansion Results Arithmetic sum Asymptotic distribution Construction of normal numbers 31 / 70

36 Fourier transform of the indicator function In the one dimensional case we can easily do Fourier transform. n N ( I l,a (p(n)) 1 ) q N δ + ( δ min ν, 1 ) ( ) ν e p(n) 2 ν ql+1. ν=1 n N 32 / 70

37 Fourier transform of ψ In the same manner we get for the function ψ ψ(g(n)) N δ + ( δ min ν, 1 ) e (νg(n)) 2 ν. n N ν=1 Recall that I l,a (x) = 1 ( x q + ψ q a + 1 ) l+1 q n N ( x ψ q a ) l+1 q 33 / 70

38 Higher dimension: Embedding Since it is more easy to consider the properties of number systems in R n, we define an embedding of K in R n. Thus let φ be defined by { φ : C R n, α 0 + α 1 b + + α n 1 b n 1 (α 0,..., α n 1 ). 34 / 70

39 Higher dimension: Matrix number system Let m b (x) = a 0 + a 1 x + + a n 1 x n 1 be the minimal polynomial of b, then we define the corresponding matrix B by 0 0 a B = a n 1 One easily checks that φ(b z) = B φ(z). 35 / 70

40 Higher dimension: Fundamental domain By this we define the embedding of the fundamental domain by { F = φ(f) = z R n z = } B k a k, a k φ(n ). k 1 36 / 70

41 Higher dimension: Twin dragon Figure: Fundamental domain for b = 1 + i 37 / 70

42 Higher dimension: Urysohn function Let F a be the domain consisting of all elements of F starting with a, i.e., F a = B 1 (F + a) In order to count the number of hits in F a we define 1 if (x 1,..., x n ) I k,a 1 ψ a (x 1,..., x n ) = if (x 2 1,..., x n ) Π k,a 0 otherwise. 38 / 70

43 Higher dimension: Urysohn function Let F a be the domain consisting of all elements of F starting with a, i.e., F a = B 1 (F + a) In order to count the number of hits in F a we define 1 if (x 1,..., x n ) I k,a 1 ψ a (x 1,..., x n ) = if (x 2 1,..., x n ) Π k,a 0 otherwise. 38 / 70

44 Higher dimension: Urysohn function Now we smooth the function ψ in order to get u a (x 1,..., x n ) = 1 n ψ a (x 1 + y 1,..., x n + y n ) dy 1 dy n, where := 2c u b k 39 / 70

45 Higher dimension: Fourier transform Finally we do Fourier transformation of the Urysohn function. Thus u a (x 1,..., x n ) = c m1,...,m n e(m 1 x m n x n ) (m 1,...,m n) Z n 40 / 70

46 Higher dimension: Estimation of Θ Now we get that Θ z P(T ) t(φ(p(z))) h F li, i=1 where t is defined by t(ν) = ( h ) T M e µ j B l j 1 ν M M with T M = h j=1 c m j1,...,m jn. j=1 41 / 70

47 Higher dimension: The border We collect all points near the border of the fundamental domain in the following set. { ( ) p(z) F l := # z P(T ) φ } P k,a mod B 1 Z n. b l+1 a N 42 / 70

48 Higher dimension: Erdős-Turan-Koksma Lemma Let x 1,..., x L be points in the n-dimensional real vector space R n and H an arbitrary positive integer. Then the discrepancy D L (x 1,..., x L ) fulfils the inequality D L (x 1,..., x L ) 2 H h H 1 r(h) where h Z n and r(h) = n i=1 max(1, h i ). 1 L L e(h x l ), l=1 43 / 70

49 Outline Motivation Number systems and additive functions Distributional properties Normal numbers Method Transfer of the problems Fourier transform Weyl Sums Division of the expansion Results Arithmetic sum Asymptotic distribution Construction of normal numbers 44 / 70

50 Weyl Sums Since in most of the examples above we used polynomials we write p(x) = α d x d + + α 1 x + α 0. Then we are interested in the estimation of sums of the following kind. ( ) ν d e α q l+1 k x k n N k=0 45 / 70

51 Higher Dimension: Weyl sums In both cases, the Fourier-Transform of the Urysohn-function and the Erdős-Turan-Koksma inequality, we end up with sums of the form e(m 1 x m n x n ). l L A typical Weyl sum in this field looks like e (Tr (g(z))). z P(T ) 46 / 70

52 Higher Dimension: Weyl sums Our sums are of the form z P(T ) e ( h φ ( p(z) b l+1 )). We can rewrite these sums and get the following ( ( )) q(z) e Tr. z P(T ) In order to estimate these Weyl sums we consider them according to the size of l. This is the reason why we do not get an estimate for the whole range of ls. b l 47 / 70

53 Higher Dimension: Siegel s Lemma Lemma Siegel Let N > D 1 n. Then, corresponding to any ξ K, there exist q O K and a δ 1 such that and max q (i) ξ (i) a (i) 1 < N, 1 i n 0 < max q (i) N, 1 i n max(n q (i) ξ (i) a (i), q (i) ) D 1 2, 1 i n, N((q, aδ)) D / 70

54 Higher Dimension: Weyl s Lemma Lemma Let g(z) = α k z k + + α 1 z with max a (i) T k (log T ) σ 1 and Then α (i) k 1 i n q(i) k (log T ) σ 1 max z P(T ) with σ 1 2 k 1 ( σ 0 + r2 2k). k q (i) k 1 i n T k (log T ) σ 1. e (Tr (g(z))) T n (log T ) σ 0 49 / 70

55 Outline Motivation Number systems and additive functions Distributional properties Normal numbers Method Transfer of the problems Fourier transform Weyl Sums Division of the expansion Results Arithmetic sum Asymptotic distribution Construction of normal numbers 50 / 70

56 Division of the expansion We will now focus on Delange s problem: f (aq l )I l,d (p(n)) n N = f (aq l ) ( N q + ( p(n) ψ q a + 1 ) ( p(n) ψ l+1 q q a ) ). l+1 q n N 51 / 70

57 Division of the expansion Since in our case the leading coefficient looks like ν q l+1 α i. The Diophantine approximation leads us to a division of the expansion according to the position of the digit within the expansion. Least significant digits (0 l < C l log log N). Middle digits (C l log log N l < C u log log N). Most significant digits (C u log log N l < log N). 52 / 70

58 Treating the middle digits By an application of Weyl s Lemma we get for C l log log N l < C u log log N that ( ) ν e p(n) N ql+1 (log N). σ n N 53 / 70

59 Treating the least significant digits This treatment depends on the shape of the coefficients of the polynomials. Under the assumption, that at least one of them is of finite type we again get that C l log log N l=0 I l,a(p(n)) 1 q n N N (log N) σ. 54 / 70

60 Treating the most significant digits Here we want to extract the periodic error part in the shape of Delange (periodic error term). Therefore we rewrite the sum as follows. N<n 2N ( p(n) ψ q a ) l+1 q = 1 1 d α d 0 q r+1 d q r 1 p(2n) q r 1 p(n) where C u log log N l < log N. ( ψ t a ) t 1 d 1 dt + O ( N 1 ε), q 55 / 70

61 Outline Motivation Number systems and additive functions Distributional properties Normal numbers Method Transfer of the problems Fourier transform Weyl Sums Division of the expansion Results Arithmetic sum Asymptotic distribution Construction of normal numbers 56 / 70

62 Peter s Result Theorem Peter (2002) There are c R and ε > 0 such that n N s q ( αn k ) = q 1 N log 2 q (αn k ) + cn + NF ( log q (αn k ) ) + O(N 1 ε ) where x is the greatest integer less than x, F a 1-periodic function and α = 1 or α > 0 an irrational of finite type. 57 / 70

63 Pseudo polynomial Let α 0, β 0,..., α d, β d R, α 0 > 0 and β 0 > β 1 > > β d 1, where at least one β i Z. Then we define a pseudo polynomial p as p(x) := α 0 x β α d x β d. 58 / 70

64 Over a pseudo-polynomial sequence Theorem Nakai and Shiokawa (1990) Let p be a pseudo polynomial. Then n N s q ( p(n) ) = q 1 N log 2 q p(n) + O(N) where log q denotes the logarithm to base q. 59 / 70

65 Arbitrary additive functions Theorem M (201?) Let q N \ {1} and f be a strictly q-additive function with f (0) = 0. If p is a pseudo polynomial, then there exists ε > 0 such that f ( p(n) ) = µ f N log q (p(n)) n N + NF ( log q (p(n)) ) + O ( N 1 ε). 60 / 70

66 Outline Motivation Number systems and additive functions Distributional properties Normal numbers Method Transfer of the problems Fourier transform Weyl Sums Division of the expansion Results Arithmetic sum Asymptotic distribution Construction of normal numbers 61 / 70

67 Length of expansion We note that with one expansion for γ you also have one for all the conjugates, by γ (i) = h a k (b (i) ) k k=0 (1 i n). Theorem Kovacs and Pethő (1992) Let l(γ) be the length of the expansion of γ to the base b. Then log γ (i) l(γ) max 1 i n log b (i) C. 62 / 70

68 Length of expansion We note that with one expansion for γ you also have one for all the conjugates, by γ (i) = h a k (b (i) ) k k=0 (1 i n). Theorem Kovacs and Pethő (1992) Let l(γ) be the length of the expansion of γ to the base b. Then log γ (i) l(γ) max 1 i n log b (i) C. 62 / 70

69 Area of interest We fix a T and set T i for 1 i n such that log T i = log T log b (i) n log N(b). Furthermore we will write N(T) = N(T 1,..., T n ) := { λ R : λ (i) Ti, 1 i n }. 63 / 70

70 Asymptotic distribution in Z[β] Theorem M (2009) Assume that there exists an ε > 0 such that D b (x)/(log x) ε as x and let p be a polynomial of degree d. Then, as T, { 1 #N(T) # z N(T) f ( p(z) ) M b (T d ) D b (T d ) where Φ is the normal distribution function. } < y Φ(y), 64 / 70

71 Some remarks It should suffices that D b (x) for x. One can shift the decimal dot. 65 / 70

72 Asymptotic distribution in function fields Let (B, N ) be a number system in S := F q [X, Y ]/pf q [X, Y ] with d(b) = a and let f : S R be a strict B-additive b function. Set µ f := 1 #N D N f (A) and σ 2 f := 1 #N f (A) 2 µ 2 f. D N 66 / 70

73 Asymptotic distribution in function fields Theorem M and Thuswaldner (2009) Let h L [Z] be a polynomial of degree d. Suppose that σ f > 0 and S is the ring of integers of L, then for n ndb f (h(a)) a # A S(n) : µ f x ndb a σ f Φ(x), where Φ denotes the standard normal distribution function. 67 / 70

74 Outline Motivation Number systems and additive functions Distributional properties Normal numbers Method Transfer of the problems Fourier transform Weyl Sums Division of the expansion Results Arithmetic sum Asymptotic distribution Construction of normal numbers 68 / 70

75 Construction of normal numbers Theorem Nakai and Shiokawa (1992) Let f be a polynomial with real coefficients. Then θ((f (n)) n 1 ) is normal. Theorem M, Thuswaldner and Tichy (2007) Let f be an entire function of bounded logarithmic order. Then θ((f (n)) n 1 ) and θ((f (p)) p P ) are normal. 69 / 70

76 Further projects Extend the existing results to the more general case of canonical number systems in algebraic number fields and function fields. Build some connections to ergodic, measure theoretic and/or analytic methods. Consider different number systems such as Cantor and Zeckendorf. 70 / 70