Distribution results for additive functions

Transcription

1 Distribution results for additive functions Manfred G. Madritsch Institute of Statistics Graz University of Technology Universiteit Stellenbosch Supported by the Austrian Research Fund (FWF), Project S / 38

2 Outline Number Systems and additive functions Distribution results Sketch of Proof 2 / 38

3 Number system A number system provides a way to describe the elements within a ring. For example for a fixed negative integer q 2 we can represent every n Z by n = a k q k + + a 1 q + a 0 where a i {0, 1,..., q 1} for i = 0, 1,..., k. 3 / 38

4 Uniqueness We call a representation of n Z unique if n = a k q k + + a 1 q + a 0 = b l q l + + b 1 q + b 0 with a i, b i {0, 1,..., q 1} implies that l = k and b i = a i for i = 1,..., k. 4 / 38

5 Finiteness We call a representation of n Z finite if n = a k q k + + a 1 q + a 0 where a i {0, 1,..., q 1} implies that there exists an integer k 0 0 such that a i = 0 for i k 0. We call k 0 the length of the expansion denoted by l(n). 5 / 38

6 Number system We set N := {0, 1,..., q 1}. Then we call (q, N ) a number system of Z if every element n Z has a unique and finite expansion. Furthermore we call q the base of this number system and N its set of digits. 6 / 38

7 Number systems in algebraic number fields A canonical step is to extend the above idea to algebraic number fields. Therefore let K be an algebraic number field of degree n and O its ring of integers. Then for b O we set D := {0, 1,..., N(b) 1} the set of digits. We call (b, D) a canonical number system if every g O has a unique and finite representation of the form whit a i D. g = a k b k + + a 1 b + a 0 7 / 38

8 Additive function Let (b, D) be a number system. Then we call a function O G, where G is a group, b-additive if ( ) f a i b i = f i (b i ). i 0 i 0 Furthermore f is called strictly b-additive if f i = f j for all i, j. An easy example is the sum of digits function s b defined by ( ) s q a i b i = a i. i 0 i 0 8 / 38

9 Outline Number Systems and additive functions Distribution results Sketch of Proof 9 / 38

10 Mean We are interested in distributional results of additive functions. One of the first was Delange, who computed the average of the sum-of-digits function, i.e., 1 N n N s q (n) = q 1 2 log q (N) + γ 1 (log q (N)), where log q denotes the logarithm in base q and γ 1 is a continuous function of period / 38

11 Distribution result for integers Theorem Bassily and Katai Let f be a q-additive function such that f (cq k ) = O(1) as k and c N. Furthermore let m k,q := 1 f (cq k ), q c N σk,q 2 := 1 f 2 (cq k ) m 2 q k,q, c N and M q (x) := with N = [log q x]. N m k,q, Dq(x) 2 = k=0 N k=0 σ 2 k,q 11 / 38

12 Distribution result for integers Theorem Bassily and Katai 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), 12 / 38

13 Distribution result for Gaussian integers Theorem Gittenberger and Thuswaldner Let f be a b-additive function such that f (cb k ) = O(1) as k and c D. Furthermore let m k,b := 1 N(b) c N f (cb k ), σ 2 k,b := 1 N(b) f 2 (cb k ) mk,b, 2 c 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 13 / 38

14 Distribution result for Gaussian integers Theorem Gittenberger and Thuswaldner 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), 14 / 38

15 Generalisation of distribution result Theorem Let f be a b-additive function such that f (cb k ) = O(1) as k and c D. Furthermore let m k,b := 1 N(b) c N f (cb k ), σ 2 k,b := 1 N(b) f 2 (cb k ) mk,b, 2 c 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 15 / 38

16 Length estimations for canonical number systems Lemma Kovács and Pethő 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. Therefore we define for an integer T P(T ) := { γ K 0 γ (l) T, 0 γ (m) T }, where l and m go through the real and imaginary conjugates, respectively. 16 / 38

17 Generalisation of distribution result Theorem Assume that D b (x)/(log x) 1/3 as x and let p O[X ] be a polynomial of degree d. Then, as T, { 1 #P(T ) # z P(T ) f (p(z)) M b (T d ) D b (T d ) where Φ is the normal distribution function. } < y Φ(y), 17 / 38

18 Outline Number Systems and additive functions Distribution results Sketch of Proof 18 / 38

19 Idea of Proof We need the following tools: Transferation to a moment estimation. Showing the right distribution of the moments. Urysohn-function for the fundamental domain. The Erdős-Turan-Koksma inequality for the border. Weyl sums. 19 / 38

20 Transferation of the problem 1. Truncate the range of the function f. 2. Consideration of the moments. 3. Getting rid of the polynomial. 20 / 38

21 Transferation: Truncation Let 0 A B L be arbitrarily chosen. Then define the truncated function f to be f (p(z)) = B f (a j (p(z))b j ). j=a We easily get that max f (p(z)) M(T d ) z P(T ) D(T d ) f (p(z)) M (T d ) D (T d ) / 38

22 Transferation: Moments Thus it suffices to consider { 1 #P(T ) # z P(T ) f (p(z)) M (T d ) D (T d ) } < y Φ(y). and by the Frechet-Shohat Theorem this holds if and only if the moments 1 ( ) f (p(z)) M (T d k ) ξ k (T ) := #P(T ) D (T d ) z P(T ) converge to the moments of the normal law. 22 / 38

23 Transferation: Polynomial Finally we want to get rid of the polynomial p and this is done by considering the following moments η k (T ) := 1 #P(T ) z P(T ) Now we are left with showing that ( ) f (z) M (T d k ). D (T d ) ξ k (T ) η k (T ) 0 for T. 23 / 38

24 Distribution Lemma log T Let L = C b + C be an upper bound for the maximal length log b of the b-adic representation of z P(T ). For we have, A := C l log L l 1 < l 2 < < l h dl C u log L =: B Θ := # { z P(T ) alj (p(z)) = b j, j = 1,..., h } = (2π)r 2 N(b) h T n + O ( T n (log T ) σ 0 ) uniformly for T, where b j D are given and σ 0 is an arbitrary positive constant. 24 / 38

25 Distribution: Embedding Since it is more easy to consider the properties of number systems in R n 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 ). 25 / 38

26 Distribution: 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). 26 / 38

27 Distribution: 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(d). k 1 27 / 38

28 Twin dragon Figure: Fundamental domain for b = 1 + i 28 / 38

29 Urysohn function In order to count the number of hits in Θ 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. 29 / 38

30 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 30 / 38

31 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 31 / 38

32 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 whit T M = h j=1 c m j1,...,m jn. j=1 32 / 38

33 The border We collect all points near the border of the fundamental domain in the following set. { ( ) p(z) F j := # z P(T ) φ } P k,a mod B 1 Z n. b j+1 a N 33 / 38

34 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 34 / 38

35 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 ) 35 / 38

36 Our sums are of the form z P(T ) Weyl sums e ( h φ ( p(z) b j+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 j. This is the reason why we do not get an estimate for the whole range of js. b j 36 / 38

37 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 / 38

38 Weyl s Lemma Lemma Let g(z) = α k z k + + α 1 z with max α (i) a (i) T k (log T ) σ 1 and Then 1 i n k q(i) k (log T ) σ 1 max with σ 1 2 k 1 ( σ 0 + r2 2k). k q (i) k 1 i n T k (log T ) σ 1. S(g, T ) T n (log T ) σ 0 38 / 38