Quantitative Diophantine approximation and Fourier dimension of sets: Dirichlet non-improvable numbers versus well-approximable numbers

TL;DR

This paper establishes a quantitative inhomogeneous Khintchine-type theorem for sets supporting measures with specific Fourier decay, extending previous results, and shows the Dirichlet non-improvable set has positive Fourier dimension.

Contribution

It introduces a new Khintchine-type theorem for lacunary sequences on sets with Fourier decay and proves the positive Fourier dimension of the Dirichlet non-improvable set.

Findings

Improved Khintchine-type theorem for lacunary sequences on Fourier-decaying sets

Demonstrated positive Fourier dimension of Dirichlet non-improvable set

Extended previous results by Pollington-Velani-Zafeiropoulos-Zorin

Abstract

Let $E\subset [0,1]$ be a set that supports a probability measure $\mu$ with the property that $|\widehat{\mu}(t)|\ll (\log |t|)^{-A}$ for some constant $A>2.$ Let $\mathcal{A}=(q_n)_{n\in \N}$ be a positive, real-valued, lacunary sequence. We present a quantitative inhomogeneous Khintchine-type theorem in which the points of interest are restricted to $E$ and the denominators of the shifted fractions are restricted to $\mathcal{A}.$ Our result improves and extends a previous result in this direction obtained by Pollington-Velani-Zafeiropoulos-Zorin (2022). We also show that the Dirichlet non-improvable set VS well-approximable set is of positive Fourier dimension.