On evil Kronecker sequences and lacunary trigonometric products
Christoph Aistleitner, Roswitha Hofer, Gerhard Larcher · Annales de l’institut Fourier · 2017
An 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 measure the speed of convergence of the empirical distribution of ( { n 1 α } , ⋯ , { n N α } ) towards the uniform distribution. In this paper we investigate the case when ( n k ) k ≥ 1 is the sequence of evil numbers, that is the sequence of non-negative integers having an even sum of digits in base 2. We utilize a connection with lacunary trigonometric products ∏ ℓ = 0 L sin π 2 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 provide some explicit examples of numbers α for which we can give estimates for the discrepancy of n k α k ≥ 1 .