On the Exact Fourier Dimension of Sets of Well-Approximable Matrices

TL;DR

This paper determines the precise Fourier dimension of sets of well-approximable matrices and numbers, extending understanding in Diophantine approximation and harmonic analysis for various approximation functions.

Contribution

It provides an exact computation of Fourier dimensions for well-approximable matrices and numbers under broad conditions, filling a gap in the literature.

Findings

Exact Fourier dimension computed for well-approximable matrices and numbers.

Results hold for any approximation function with summable $igl( extstyle ext{sum over } qigr)$.

Extends previous work to both homogeneous and inhomogeneous cases.

Abstract

We compute the exact Fourier dimension of the set of $\Psi$-well-approximable $m \times n$ matrices (and the set of $\Psi$-well-approximable numbers) in the homogeneous and inhomogeneous cases for any approximation function $\Psi$ satisfying $\sum_{q \in \mathbb{Z}^n} \Psi(q)^m < \infty$.