Non-Salem sets in multiplicative Diophantine approximation

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 $\tau>1.$