Sharp Fourier decay estimates for measures supported on the well-approximable numbers

TL;DR

This paper constructs a measure supported on well-approximable numbers with nearly optimal Fourier decay, improving previous results and advancing understanding of harmonic analysis on these sets.

Contribution

It introduces a new measure with enhanced Fourier decay properties on well-approximable numbers, surpassing earlier constructions.

Findings

Fourier transform decays at a nearly optimal rate

Provides a logarithmic improvement over Kaufman's construction

Advances harmonic analysis on well-approximable sets

Abstract

We construct a measure on the well-approximable numbers whose Fourier transform decays at a nearly optimal rate. This gives a logarithmic improvement on a previous construction of Kaufman.