Uniform distribution of sequences connected with the weighted sum-of-digits function

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1 Uniform distribution of seuences connected with the weighted sum-of-digits function Friedrich Pillichshammer Abstract In this paper we consider seuences which are connected with the so-called weighted -ary sum-of-digits function and give an if and only if condition under which such seuences are uniformly distributed modulo one. The seuences considered here contain the -ary van der Corput seuence as well as the (nα-seuences as special cases. AMS subject classification: 11K06, 11J71. 1 Introduction A seuence (x n n 0 in the d-dimensional unit-cube is said to be uniformly distributed modulo one if for all intervals [a, b [0, 1 d we have #{n : 0 n < N, x n [a, b} lim = λ d ([a, b, N N where λ d denotes the d-dimensional Lebesgue measure. An excellent introduction into this topic can be found in the book of Kuipers and Niederreiter [7] or in the book of Drmota and Tichy [4]. In this paper we consider the uniform distribution properties of special seuences which are connected with the weighted sum-of-digits function and which are generalizations of many well known seuences. Let γ = (γ 0, γ 1,... be a seuence in R and let N, 2. For n N 0 with base representation n = n 0 + n 1 + n we define the weighted -ary sum-of-digits function by s γ (n := γ 0 n 0 + γ 1 n 1 + γ 2 n 2 +. We remark that the weighted -ary sum-of-digits function is a -additive function, but it is not strongly -additive (unless the weight-seuence γ is constant; see [5, 6] or [4] for the notion of (strongly -additive functions. For d N let γ = (γ 0, γ 1,... be a seuences in R d with γ j = (γ (1 j,..., γ (d j, i.e., γ (k j denotes the k-th component of the j-th element of the seuence γ. For k {1,..., d} let γ (k = (γ (k 0, γ(k 1,... be the k-th coordinate seuence in R. For n N 0 define s γ (n := (s γ (1(n,..., s γ (d(n. This work is supported by the Austrian Science Foundation (FWF, Project S9609, that is part of the Austrian National Research Network Analytic Combinatorics and Probabilistic Number Theory. 1

2 Now we consider the d-dimensional seuence ({s γ (n} n 0, (1 where {x} denotes the fractional part of the vector x (applied component-wise, and ask under which conditions on the weight-seuence γ the seuence (1 is uniformly distributed modulo one? Observe that the definition of the seuence in (1 covers many well known and extensively studied seuences as, for example: 1. If d = 1 and γ j = (here we simply write γ j instead of γ (1 j for all j N 0, then the seuence ({s γ (n} n 0 is the -ary van der Corput seuence which is of course well known to be uniformly distributed modulo one. See, for example, [7, 11]. 2. If γ j = j α for all j N 0 with α = (α 1,..., α d R d, then we obtain the seuence ({nα} n 1 which is well known to be uniformly distributed modulo one if and only if 1, α 1,..., α d are linearly independent over Q. See, for example, [4, 7, 12]. 3. If γ j = α = (α 1,..., α d R d for all j N 0, then we obtain the seuence ({s(nα} n 1, where s( denotes the classical, i.e. unweighted -ary sum-of-digits function. In the case d = 1 it was shown by Mendès France [10] and later by Couet [1] that the seuence ({s(nα} n 1 is uniformly distributed modulo one if and only if α R \ Q. See also [2, 3] and the references therein. We remark that this result even holds if the -ary sum-of-digits function is replaced by an arbitrary strongly -additive function; see [4]. 4. If d = 1 and γ j = r j α (again we simply write γ j instead of γ (1 j with r j Z for all j N 0 where α R, then the following was proved (in fact in a more general setting by Larcher [8]: the seuence ({s γ (n} n 0 is uniformly distributed modulo one if and only if hr k α 2 = h N, where for x R, x = min k Z x k. It is the aim of this paper to characterize the weight-seuences γ : N 0 R d for which the seuence (1 is uniformly distributed modulo one. As corollary we obtain that the seuence (1 is uniformly distributed modulo one for almost all weight-seuences γ : N 0 [0, 1 d. We close the paper with an interesting open uestion. Throughout the paper let the base N, 2, and the dimension d N be fixed. By, we denote the usual inner product in R d. As above denotes the distance-tothe-nearest-integer function. 2 Statement and proof of the results The following theorem gives a full characterization of the seuences γ : N 0 R d for which the seuence (1 is uniformly distributed modulo one. The proof is based on easy estimates for exponential sums and Weyl s criterion (see, for example, [4, 7]. 2

3 Theorem 1 The seuence ({s γ (n} n 0 is uniformly distributed modulo one if and only if for every h Z d \ {0} one of the following properties hold: Either h, γ k 2 = or there exists a k N 0 such that h, γ k Z and h, γ k Z. Of course the condition from our theorem covers all special cases from the list of examples in Section 1. Before we give the proof of the theorem let us consider two of them. Example 1 Consider the -ary van der Corput seuence, i.e., d = 1 and γ j = for all j N 0. For h Z \ {0} let k N 0 be maximal such that k h. Then h k 1 Z and h k Z. Hence from Theorem 1 we obtain the well known fact that the -ary van der Corput seuence is uniformly distributed modulo one. Example 2 Let γ j = α = (α 1,..., α d R d for all j N 0. Then for any h Z d \ {0} we have h, γ k 2 = h, α 2 = if and only if h, α Z. But the last condition holds if and only if 1, α 1,..., α d are linearly independent over Q. For the proof of Theorem 1 we need the following easy lemmas. completeness we give short verifications of these results. For the sake of Lemma 1 Let x 0,..., x 1 ( 1 2, 1 2] and define x := max0j< x j. Then we have 1 e 2πix j (1 4π2 x 2. Proof. We have 1 e 2πix j Re 1 e j 2πix = cos(2πx j cos(2πx (1 4π2 x 2. ( 1 Lemma 2 For any x R we have 1 e 2πixn 4 x 2. 3

4 Proof. We have 1 e 2πixn 1 + e 2πix + 2 = 2 cos(π x (1 π2 x x 2. π Proof of Theorem 1. Let h Z d \ {0}. By Lemma 2 we have 1 e 2πi h,γ k n 4 h, γ k 2. But if h, γ k Z and h, γ k Z we also have 1 e 2πi h,γ k n = 0. For j N 0 we have 1 e 2πi h,sγ(n j = 1 1 e 2πi h,γ j k n 4 h, γ k 2 h,γ k Z h,γ k Z Here and later on an empty product is considered to be one. Let N N with base representation N = N 0 + N N m m with N m 0. For 0 j m set N(j := N j j + + N m m. Define g(n := e 2πi h,sγ(n. Then Now and N(m 1 N( n=n(j+1 Therefore N 1 e 2πi h,sγ(n N 1 g(n = e 2πi h,sγ(n = N m 1 l=0 (l+1 m 1 N(m 1 N j g(n = g(n(j + 1 g(n + m 1 n=l m e 2πi h,n0γ0+ +nmγm = N( n=n(j+1 N m 1 l=0 N g(n = g(n(j + 1 g(n. m 1 g(l m l=0 N m g(l j l=0 g(n m N j j 1 j g(n r 1 m ( 4 h, N j j + N j j γk 2 j=r 4 g(l j g(n, g(n. h,γ k Z h,γ k Z 0. 0.

5 for any r N 0. We consider two cases 1. There exists a k N 0 such that h, γ k Z and h, γ k Z. Let k 0 be minimal with this property (of course k 0 is independent of N. Then we have N 1 k 0 e 2πi h,sγ(n N j j ( 4 h, γk 2 k 0 ( 1 j = k For all k N 0 we have h, γ k Z or h, γ k Z. Then we have N 1 r 1 ( 4 h, e 2πi h,sγ(n r γk 2 + N. (2 Define x r := r / = r 1 r 1 h,γ k Z ( 4 h, γk 2 r 1 ( 2 4 h, γ k 2 r. Therefore x r as r. Choose r such that x r N < x r+1. Then we have r N r 1 On the other hand we have r ( 4 h, γk 2 ( 4 h, γk 2 r ( 4 h, γk 2. (3 r 1 = 1 r+1 and hence N < r+1 / r ( 4 h, γk 2 2(r+1. Thus we have log N < r + 1 resp. log N r and hence r 1 ( 4 h, γk 2 log N 1 ( 4 h, γk 2. (4 5

6 From (2, (3 and (4 we find N 1 e 2πi h,sγ(n 2N 2Ne 2Ne log N 1 log N 1 P 4 ( 4 h, γk 2 log log N 1 P «4 h,γ k 2 h,γ k 2 In both of the above cases we obtain 1 N 1 N e2πi h,sγ(n 0 as N. Hence the result follows by Weyl s criterion. Assume now that there is a h Z d \ {0} such that h, γ k 2 < and for all k N 0 we have h, γ k Z or h, γ k Z. Then we have h, γ k 2 = h, γ k 2 + h,γ k Z h, γ k 2 <. For j N 0 we have 1 j e 2πi h,sγ(n = 1 j 1 e 2πi h,γ k n. Here we have 1 e 2πi h,γ k n 0 for all k N 0. This is clear for the case h, γ k Z. If h, γ k Z, then we have h, γ k Z and the ineuality holds as well. With Lemma 1 and since nx n x for all n N 0 we obtain 1 ( e 2πi h,γ k n 1 4π 2 max h, γ k n 2 > ( 1 4π 2 h, γ k 2. 0n< Let 0 < c < 1 and let l N be large enough such that 1 4π 2 k>l h, γ k 2 > c > 0. 6

7 For j > l we have 1 j e 2πi h,sγ(n l 1 1 j ( e 2πi h,γ k n 1 4π 2 h, γ k 2 k=l+1 ( c 1 4π 2 h, γ k 2 > c c > 0. k>l and by Weyl s criterion ({s γ (n} n 0 is not uniformly distributed modulo one. Corollary 1 The seuence ({s γ (n} n 0 is uniformly distributed modulo one for almost all seuences γ : N 0 [0, 1 d. Proof. We consider the seuence of random variables X 1, X 2,... uniformly i.i.d. in [0, 1 d. For h Z d \ {0}, we have E( h, X i 2 = 1/12 and hence it follows from Kolmogorov s strong law of large numbers that for n we have Therefore h, X h, X n 2 n 1 12 h, γ k 2 = for almost all seuences γ : N 0 [0, 1 d and hence a.e.. h, γ k 2 = h Z d \ {0} for almost all seuences γ : N 0 [0, 1 d. The result follows from Theorem 1. Finally we state an Open uestion: Let 1,..., d 2 be pairwisely coprime integers. Under which conditions on the weight-seuences γ (k = (γ (k,... in R, k {1,..., d}, is the seuence 0, γ(k 1 ({(s 1,γ (1(n,..., s 1,γ (1(n} n 0 (5 uniformly distributed modulo one? (Here we wrote s,γ ( for the weighted -ary sum-ofdigits function to stress the dependence on the base. For example if γ (k i = i 1 k for all k {1,..., d} and all i N 0, then we obtain the d-dimensional Halton seuences which is well known to be uniformly distributed modulo one. If γ (k i = α k R for all k {1,..., d} and all i N 0, then it was shown by Drmota and Larcher [3] that the seuence (5 is uniformly distributed modulo one if and only if α 1,..., α d R \ Q. But also the classical (nα-seuence is contained in this concept. 7

8 References [1] J. Couet, Sur certaines suites uniformément éuiréparties modulo 1. Acta Arith. 36 (1980, [2] M. Drmota, -additive functions and well distribution. Demonstratio Math. 30 (1997, [3] M. Drmota, G. Larcher, The sum-of-digits function and uniform distribution modulo 1. J. Number Theory 89 (2001, [4] M. Drmota, R.F. Tichy, Seuences, Discrepancies and Applications. Lecture Notes in Mathematics 1651, Springer-Verlag, Berlin, [5] Gel fond, A.O., Sur les nombres ui ont des propriétés additives et multiplicatives données. Acta Arith. 13 (1968, [6] Kátai, I., On -additive and -multiplicative functions, In: Number theory and discrete mathematics, (A.K. Agarwal, B.C. Berndt, C.F. Krattenthaler, G.L. Mullen, K. Ramachandra and M. Waldschmidt, eds., (Chandigarh, 2000, Trends in Math., Birkhäuser, Basel, pp , [7] L. Kuipers, H. Niederreiter, Uniform Distribution of Seuences. John Wiley, New York, [8] G. Larcher, On the distribution of seuences connected with digit-representation. Manuscripta Math. 61 (1988, [9] G. Larcher, R.F. Tichy, Some number-theoretical properties of generalized sum-ofdigits functions. Acta Arith. 52 (1989, [10] M. Mendès France, Nombres normaux, applications aux fonctions pseudoaléatories. J. Anal. Math. 20 (1967, [11] P.D. Proinov, V.S. Grozdanov, On the diaphony of the van der Corput-Halton Seuence. J. Number Theory 30 (1988, [12] O. Strauch, Š. Porubský, Distribution of Seuences: A Sampler. Peter Lang, Bern, Friedrich Pillichshammer, Institut für Finanzmathematik, Universität Linz, Altenbergstraße 69, A-4040 Linz, Austria. friedrich.pillichshammer@jku.at 8