The dispersion of dilated lacunary sequences, with applications in multiplicative Diophantine approximation

TL;DR

This paper investigates the distribution of dilated lacunary sequences modulo 1, providing bounds on maximal gaps, and applies these findings to improve results in multiplicative Diophantine approximation.

Contribution

It establishes new upper bounds for gaps in dilated lacunary sequences and applies these to enhance understanding in multiplicative Diophantine approximation.

Findings

Maximal gaps are bounded by (log N)/N for some dilations.

For almost all dilations, gaps are bounded by (log N)^{2+ε}/N.

Results improve previous bounds and are nearly optimal.

Abstract

Let $(a_n)_{n \in \mathbb{N}}$ be a Hadamard lacunary sequence. We give upper bounds for the maximal gap of the set of dilates $\{a_n \alpha\}_{n \leq N}$ modulo 1, in terms of $N$. For any lacunary sequence $(a_n)_{n \in \mathbb{N}}$ we prove the existence of a dilation factor $\alpha$ such that the maximal gap is of order at most $(\log N)/N$, and we prove that for Lebesgue almost all $\alpha$ the maximal gap is of order at most $(\log N)^{2+\varepsilon}/N$. The metric result is generalized to other measures satisfying a certain Fourier decay assumption. Both upper bounds are optimal up to a factor of logarithmic order, and the latter result improves a recent result of Chow and Technau. Finally, we show that our result implies an improved upper bound in the inhomogeneous version of Littlewood's problem in multiplicative Diophantine approximation.