Fourier Dimension Estimates for Sets of Exact Approximation Order: The Badly-Approximable Case

TL;DR

This paper proves that for certain approximation functions, the set of numbers exactly approximable at that rate has positive Fourier dimension, implying it contains normal numbers, thus linking approximation properties with harmonic analysis.

Contribution

It establishes positive Fourier dimension for sets of exactly approximable numbers under specific decay and limit conditions on the approximation function.

Findings

The set has positive Fourier dimension under given conditions.

The set contains normal numbers.

Fourier dimension relates to the approximation rate.

Abstract

We show for decreasing, positive approximation functions $\psi$ such that $\tau = \lim_{q \to \infty} \frac{\log \psi(q)}{\log q} < \frac{13 + \sqrt{73}}{8}$ and such that $q^2 \psi(q) \to 0$ that the set $\text{Exact}(\psi)$ of numbers approximable to the exact order $\psi$ has positive Fourier dimension. This implies that the set $\text{Exact}(\psi)$ contains normal numbers.