On parametric Thue-Morse Sequences and Lacunary Trigonometric Products

One of the fundamental theorems of uniform distribution theory states that the fractional parts of the sequence $(n \alpha)_{n \geq 1}$ are uniformly distributed modulo one (u.d. mod 1) for every irrational number $\alpha$. Another important result of Weyl states that for every sequence $(n_k)_{k \geq 1}$ of distinct positive integers the sequence of fractional parts of $(n_k \alpha)_{k \geq 1}$ is u.d. mod 1 for almost all $\alpha$. However, in this general case it is usually extremely difficult to classify those $\alpha$ for which uniform distribution occurs, and to measure the speed of convergence of the empirical distribution of $(\{n_1 \alpha\}, ..., \{n_N \alpha\})$ towards the uniform distribution. In the present paper we investigate this problem in the case when $(n_k)_{k \geq 1}$ 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 $\prod^{L}_{\ell=0} |\sin \pi 2^{\ell} \alpha |$, and by giving sharp metric estimates for such products we derive sharp metric estimates for exponential sums of $(n_{k} \alpha)_{k \geq 1}$ and for the discrepancy of $(\{n_{k} \alpha\})_{k \geq 1}.$ 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 $\alpha$ for which we can give estimates for the discrepancy of $(\{n_{k} \alpha\})_{k \geq1}$.