On the distribution of sequences of the form $(q_ny)$

TL;DR

This paper investigates the distribution properties of sequences formed by multiplying increasing integer sequences with a real number, providing bounds on measure and Hausdorff dimension, with applications to approximation problems.

Contribution

It offers new bounds on Hausdorff dimension and measure for well-approximated points in such sequences, especially for sequences with rapid growth and measures of positive Fourier dimension.

Findings

Bounds on Hausdorff dimension are sharp for rapidly growing sequences.

Measure bounds are sharp when the measure has positive Fourier dimension.

Applications to inhomogeneous Littlewood type approximation problems.

Abstract

We study the distribution of sequences of the form $(q_ny)_{n=1}^\infty$, where $(q_n)_{n=1}^\infty$ is some increasing sequence of integers. In particular, we study the Lebesgue measure and find bounds on the Hausdorff dimension of the set of points $\gamma \in [0,1)$ which are well approximated by points in the sequence $(q_ny)_{n=1}^\infty$. The bounds on Hausdorff dimension are valid for almost every $y$ in the support of a measure of positive Fourier dimension. When the required rate of approximation is very good or if our sequence is sufficiently rapidly growing, our dimension bounds are sharp. If the measure of positive Fourier dimension is itself Lebesgue measure, our measure bounds are also sharp for a very large class of sequences. We also give an application to inhomogeneous Littlewood type problems.