On evil Kronecker sequences and lacunary trigonometric products

Abstract

One of the fundamental theorems of uniform distribution theory states that the fractional parts of the sequence (nα) n≥1 are uniformly distributed modulo one (u.d. mod 1) for every irrational number α. Another important result of Weyl states that for every sequence (n k ) k≥1 of distinct positive integers the sequence of fractional parts of (n k α) k≥1 is u.d. mod 1 for almost all α. However, in this general case it is usually extremely difficult to classify those α for which uniform distribution occurs, and to measure the speed of convergence of the empirical distribution of ({n 1 α}, . . . , {n N α}) towards the uniform distribution. In the present paper we investigate this problem in the case when (n k ) k≥1 is the Thue-Morse sequence of integers, which means the sequence of positive integers having an even sum of digits in base 2. In particular we utilize a connection with lacunary trigonometric products L ℓ=0 sin π2 ℓ α , and by giving sharp metric estimates for such products we derive sharp metric estimates for exponential sums of (n k α) k≥1 and for the discrepancy of ({n k α}) k≥1 . Furthermore, we comment on the connection between our results and an open problem in the metric theory of Diophantine approximation, and we provide some explicit examples of numbers α for which we can give estimates for the discrepancy of ({n k α}) k≥1 .

On sequences with prescribed metric discrepancy behavior

“…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…”

“…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.…”

Additive Energy and Irregularities of Distribution

“…This result should be compared to another example which was given in [3]. There the sequence (a n ) n≥1 of Thue-Morse integers (also called evil numbers) was studied.…”

7. On pair correlation of sequences

“…Recently, a much more general metric result on PPC of sequences of the form ({a n α}) n≥1 was given in [6] which shows that there is an intimate connection between the concept of PPC of sequences ({a n α}) n≥1 and the notion of additive energy of the sequence (a n ) n≥1 . The concept of additive energy plays a central role in additive combinatorics and also appears in the study of the metrical discrepancy theory of sequences ({a n α}) n≥1 (see [2,5]). For a strictly increasing sequence a 1 < a 2 < a 3 < .…”