On parametric Thue-Morse Sequences and Lacunary Trigonometric Products

TL;DR

This paper studies the uniform distribution of the Thue-Morse sequence modulo one, establishing sharp metric estimates for related lacunary trigonometric products and their implications for discrepancy and Diophantine approximation.

Contribution

It provides new sharp metric estimates for lacunary trigonometric products associated with the Thue-Morse sequence, linking these to uniform distribution and discrepancy results.

Findings

Sharp metric estimates for lacunary trigonometric products involving the Thue-Morse sequence

Explicit discrepancy bounds for fractional parts of the Thue-Morse sequence

Connections made between these results and open problems in Diophantine approximation

Abstract

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 $(\{n_{k} \alpha\})_{k \geq1}$.