Non-Salem sets in multiplicative Diophantine approximation

In this paper, we answer a question of Cai-Hambrook in (arXiv$\text{\hspace{0.17em}⁣}:$ 2403.19410). Furthermore, we compute the Fourier dimension of the multiplicative $\psi$-well approximable set $$M_2^{\times }(\psi )=\{(x_1,x_2)\in [0,1{]}^2\text{\hspace{0.17em}⁣}:\mathrm{\parallel }q{x}_1\mathrm{\parallel }\mathrm{\parallel }q{x}_2\mathrm{\parallel }<\psi (q)\text{ for infinitely many }q\in \mathbb{N}\},$$ where $\psi \text{\hspace{0.17em}⁣}:\mathbb{N}\to [0,\frac{1}{4})$ is a positive function satisfying ${\sum }_q\psi (q)\log \frac{1}{\psi (q)}<\mathrm{\infty }\mathrm{.}$ As a corollary, we show that the set $M_2^{\times }(q\mapsto q^{-\tau })$ is non-Salem for $\tau >1.$