Fourier Dimension Estimates for Sets of Exact Approximation Order: the Well-Approximable Case

TL;DR

This paper establishes a Fourier dimension estimate for sets of numbers with a specific approximation property, showing these sets include normal numbers, thus linking approximation theory with normality in number theory.

Contribution

It provides the first Fourier dimension estimate for sets of exact approximation order, connecting approximation properties with Fourier analysis and normality.

Findings

Sets of exact approximation order have positive Fourier dimension.

These sets contain normal numbers.

The results bridge approximation theory and harmonic analysis.

Abstract

We obtain a Fourier dimension estimate for sets of exact approximation order introduced by Bugeaud for certain approximation functions $\psi$. This Fourier dimension estimate implies that these sets of exact approximation order contain normal numbers.