Non-Salem sets in multiplicative Diophantine approximation

Abstract

In this paper, we answer a question of Cai-Hambrook in (arXiv$:$ 2403.19410). Furthermore, we compute the Fourier dimension of the multiplicative $\ddot{I}ˆ$-well approximable set $$M_2^{\times }(\ddot{I}ˆ)=\left\{(x_1,x_2)\in [0,1{]}^2:\Vert q{x}_1\Vert \Vert q{x}_2\Vert <\ddot{I}ˆ(q)\text{ for infinitely many }q\in \mathbb{N}\right\},$$ where $\ddot{I}ˆ:\mathbb{N}\to [0,\frac{1}{4})$ is a positive function satisfying ${\sum }_q\ddot{I}ˆ(q)\log \frac{1}{\ddot{I}ˆ(q)}<\infty .$ As a corollary, we show that the set $M_2^{\times }(q\mapsto q^{-\ddot{I}„})$ is non-Salem for $\ddot{I}„>1.$

15 pages