Non-Salem sets in multiplicative Diophantine approximation

TL;DR

This paper investigates the structure of multiplicative Diophantine approximation sets, computes their Fourier dimension, and shows that certain sets are non-Salem, answering a question posed by Cai-Hambrook.

Contribution

It provides a computation of the Fourier dimension for multiplicative approximation sets and establishes non-Salem properties for specific parameter choices, advancing understanding in multiplicative Diophantine approximation.

Findings

Computed Fourier dimension of multiplicative $ extit{ extbf{ψ}}$-well approximable sets.

Proved the set $M_2^{ imes}(q o q^{- au})$ is non-Salem for $ au>1$.

Answered a question of Cai-Hambrook regarding the structure of these sets.

Abstract

In this paper, we answer a question of Cai-Hambrook in (arXiv$\colon$ 2403.19410). Furthermore, we compute the Fourier dimension of the multiplicative $\psi$-well approximable set $$M_2^{\times}(\psi)=\left\{(x_1,x_2)\in [0,1]^{2}\colon \|qx_1\|\|qx_2\|<\psi(q) \text{ for infinitely many } q\in \N\right\},$$ where $\psi\colon\N\to [0,\frac{1}{4})$ is a positive function satisfying $\sum_q\psi(q)\log\frac{1}{\psi(q)}<\infty.$ As a corollary, we show that the set $M_2^{\times}(q\mapsto q^{-\tau})$ is non-Salem for $\tau>1.$