Twisted approximation with restricted denominators

TL;DR

This paper investigates the size of sets of real numbers approximable by twisted Diophantine conditions with restricted denominators, extending recent results and applying to measures with positive Fourier dimension.

Contribution

It provides new results on the measure-theoretic size of twisted approximation sets with restricted denominators for almost every real number, answering open questions in the field.

Findings

Results hold for almost every α with respect to measures of positive Fourier dimension.

Extends recent work of Kristensen and Persson.

Answers previously posed questions in twisted Diophantine approximation.

Abstract

Given an increasing integer sequence $(a_n)$, a real number $\alpha$, and a sequence $\psi(n)$, we study the set $W$ of real numbers $\gamma$ for which $a_n\alpha - \gamma$ is a distance less than $\psi(n)$ away from an integer. This is often referred to as twisted Diophantine approximation, in this case with denominators restricted to the given sequence $(a_n)$. Our main results are about the size of $W$, and they hold for almost every $\alpha$, with respect to a measure of positive Fourier dimension, for example Lebesgue measure. Our results extend recent work of Kristensen and Persson, and answer questions that they posed.