On sequences with prescribed metric discrepancy behavior

An important result of H. Weyl states that for every sequence (an)n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_{n})_{n\ge 1}$$\end{document} of distinct positive integers the sequence of fractional parts of (anα)n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_{n} \alpha )_{n \ge 1}$$\end{document} is uniformly distributed modulo one for almost all α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}. However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy DN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{N}$$\end{document} of ({anα})n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\{a_{n} \alpha \})_{n \ge 1}$$\end{document} for almost all α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}. By a result of R. C. Baker this discrepancy always satisfies NDN=O(N12+ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N D_{N} = \mathcal {O} (N^{\frac{1}{2}+\varepsilon })$$\end{document} for almost all α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} and all ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document}. In the present note for arbitrary γ∈(0,12]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \in (0, \frac{1}{2}]$$\end{document} we construct a sequence (an)n≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(a_{n})_{n \ge 1}$$\end{document} such that for almost all α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} we have NDN=O(Nγ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ND_{N} = \mathcal {O} (N^{\gamma })$$\end{document} and NDN=Ω(Nγ-ε)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ND_{N} = \Omega (N^{\gamma -\varepsilon })$$\end{document} for all ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}, thereby proving that any prescribed metric discrepancy behavior within the admissible range can actually be realized.


Introduction
Weyl [12] proved that for every sequence (a n ) n≥1 of distinct positive integers the sequence ({a n α}) n≥1 is uniformly distributed modulo one for almost all reals α. Here, and in the sequel, {·} denotes the fractional part function. The speed of convergence Communicated by J. Schoißengeier. B Christoph Aistleitner christoph.aistleitner@jku.at Gerhard Larcher gerhard.larcher@jku.at 1 Institute of Financial Mathematics and Applied Number Theory, University Linz, Linz, Austria towards the uniform distribution is measured in terms of the discrepancy, which-for an arbitrary sequence (x n ) n≥1 of points in [ 0, 1 )-is defined by For a given sequence (a n ) n≥1 it is usually a very hard and challenging problem to give sharp estimates for the discrepancy D N of ({a n α}) n≥1 valid for almost all α. For general background on uniform distribution theory and discrepancy theory see for example the monographs [6,9]. A famous result of Baker [3] states that for any sequence (a n ) n≥1 of distinct positive integers for the discrepancy D N of ({a n α}) n≥1 we have for almost all α and for all ε > 0. Note that (1) is a general upper bound which holds for all sequences (a n ) n≥1 ; however, for some specific sequences the precise typical order of decay of the discrepancy of ({a n α}) n≥1 can differ significantly from the upper bound in (1). The fact that (1) is essentially optimal (apart from logarithmic factors) as a general result covering all possible sequences can for example be seen by considering so-called lacunary sequences (a n ) n≥1 , i.e., sequences for which a n+1 a n ≥ 1 + δ for a fixed δ > 0 and all n large enough. In this case for D N we have for almost all α (see [10]), which shows that the exponent 1/2 of N on the righthand side of (1) cannot be reduced for this type of sequence. For more information concerning possible improvements of the logarithmic factor in (1), see [5]. Quite recently in [2] it was shown that also for a large class of sequences with polynomial growth behavior Baker's result is essentially best possible. For example, the following result was shown there: let f ∈ Z [x] be a polynomial of degree larger or equal to 2. Then for the discrepancy D N of ({ f (n)α}) n≥1 for almost all α and for all ε > 0 we have On the other hand there is the classical example of the Kronecker sequence, i.e., a n = n, which shows that the actual metric discrepancy behavior of ({a n α}) n≥1 can differ vastly from the general upper bound in (1). Namely, for the discrepancy of the sequence ({nα}) n≥1 for almost all α and for all ε > 0 we have which follows from classical results of Khintchine in the metric theory of continued fractions (for even more precise results, see [11]). The estimate (2) of course also holds for a n = f (n) with f ∈ Z [x] of degree 1. In [2] further examples for (a n ) n≥1 were given, where (a n ) n≥1 has polynomial growth behavior of arbitrary degree, such that for the discrepancy of ({a n α}) n≥1 we have for almost all α and for all ε > 0; see there for more details. These results may seduce to the hypothesis that for all choices of (a n ) n≥1 for the discrepancy of ({a n α}) n≥1 for almost all α we either have or This hypothesis, however, is wrong as was shown in [1]: let (a n ) n≥1 be the sequence of those positive integers with an even sum of digits in base 2, sorted in increasing order; that is (a n ) n≥1 = (3, 5, 6, 9, 10, . . . ). Then for the discrepancy of ({a n α}) n≥1 for almost all α we have and for all ε > 0, where κ is a constant with κ ≈ 0.404. Interestingly, the precise value of κ is unknown; see [8] for the background. The aim of the present paper is to show that the example above is not a singular counter-example, but that indeed "everything" between (3) and (4) is possible. More precisely, we will show the following theorem.
Then there exists a strictly increasing sequence (a n ) n≥1 of positive integers such that for the discrepancy of the sequence ({a n α}) n≥1 for almost all α we have for all ε > 0.

Proof of the theorem
For the proof we need an auxiliary result which easily follows from classical work of Behnke [4].

Lemma 1
Let (e k ) k≥1 be a strictly increasing sequence of positive integers. Let ε > 0. Then for almost all α there is a constant K (α, ε) > 0 such that for all r ∈ N there exist M r ≤ e r such that for the discrepancy of the sequence ({n 2 α}) n≥1 we have Proof For α ∈ R let a k (α) denote the kth continued fraction coefficient in the continued fraction expansion of α. Then it is well-known that for almost all α we have a k (α) = O k 1+ε for all ε > 0. Let ε > 0 be given and let α and c (α, ε) be such that for all k ≥ 1.
Since q l ≥ 2 l 2 in any case, we have l ≤ 2 log q l log 2 , and we obtain for an appropriate constant c 1 (α, ε). In [4] it was shown in Satz XVII that for every real α we have Indeed, if we follow the proof of this theorem we find that even the following was shown: for every α and for every best approximation denominator q l of α there exists an Y l < √ q l such that Y l n=1 e 2πin 2 α ≥ c abs √ q l . Here c abs is a positive absolute constant (not depending on α). Let now r ∈ N be given and let l be such that q l ≤ e r < q l+1 , and let M r := Y l from above. Then by (6) and (7) we obtain, for an appropriate constant c 2 (α, ε), By the fact that (see Chapter 2, Corollary 5.1 of [9]) which is a special case of Koksma's inequality, the result follows.
Now we are ready to prove the main theorem.
Proof of Theorem 1 Let (m j ) j≥1 and (e j ) j≥1 be two strictly increasing sequences of positive integers, which will be determined later. We will consider the following strictly increasing sequence of positive integers, which will be our sequence (a n ) n≥1 : 1, 2, 3, . . . , m 1 . . .

Furthermore, let
The sequence (a n ) n≥1 is constructed in such a way that it contains sections where it grows like (n) n≥1 as well as sections where it grows like (n 2 ) n≥1 . By this construction we exploit both the strong upper bounds for the discrepancy of ({nα}) n≥1 and the strong lower bounds for the discrepancy of ({n 2 α}) n≥1 , in an appropriately balanced way, in order to obtain the desired discrepancy behavior of the sequence ({a n α}) n≥1 . In our argument we will repeatedly make use of the fact that Let α be such that it satisfies (5) with ε = 1 2 . Then it is also well-known (see for example [9]) that for the discrepancy D N of the sequence ({nα}) n≥1 we have for all N ≥ 2.
By the above mentioned general result of Baker, that is by (1), we know that for almost all α for the discrepancy D N of the sequence ({n 2 α}) n≥1 we have for all ε > 0 and for all N ≥ 2, for an appropriate constant c 3 (α, ε). Actually an even slightly sharper estimate was given for the special case of the sequence ({n 2 α}) n≥1 by Fiedler et al. [7], who proved that for almost all α and for all ε > 0 and all N ≥ 2.
Assume that α satisfies (10) with ε = 1 8 . Then for all N ≥ 2. Now for such α and for arbitrary N we consider the discrepancy D N of the sequence ({a n α}) n≥1 .
where D x,y denotes the discrepancy of the point set ({a n α}) n=x+1,x+2,...,y . Hence by (8), (9) and by the trivial estimate D B l−1 ≤ 1 we have for all l large enough, provided that [condition (i)] m l is chosen such that (log m l ) 2 ≥ E l−1 .
Case 2 Let F l < N ≤ E l for some l. Then by Case 1 and by (8) and (11) we have for l large enough that 3 8 .
We choose [condition (ii)] Note that conditions (i) and (ii) do not depend on α. Now assume that l is so large that 2 (log F l ) 2 < F l γ 2 . Then and (note that γ ≤ 1 2 ) F l < N ≤ E l = F l + e l ≤ 2F l .
Case 3 Let E l < N < F l+1 for some l. Then by Case 2 and by (8) and (9) we have It remains to show that for every ε > 0 we have N D N ≥ N γ −ε for infinitely many N . Let l be given and let M l ≤ e l with the properties given in Lemma 1. Let N := F l + M l . Then by Lemma 1, Case 1, (8), (12) and (13) for l large enough we have This proves the theorem.