A classical result of Kaufman states that, for each $\tau>1$, the set of $\tau $-well approximable numbers $$ \begin{align*} & E(\tau)=\{x \in \mathbb{R}: |xq-r| < |q|^{-\tau} \text{ for infinitely many integer pairs } (q,r)\} \end{align*}$$is a Salem set. A natural question to ask is whether the same is true for the sets of $\tau $-well approximable $n \times d$ matrices when $nd>1$ and $\tau> d/n$. We prove the answer is no by computing the Fourier dimension of these sets. In addition, we show that the set of badly approximable $n \times d$ matrices is not Salem when $nd> 1$. The case of $nd=1$, that is, the badly approximable numbers, remains unresolved.

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度量丢番图逼近中的非塞勒姆集

Kaufman 的经典结果表明，对于每个 $\tau>1$，$\tau $-well 可逼近数的集合 $$ \begin{align*} & E(\tau)=\{x \in \mathbb {R}：|xq-r| < |q|^{-\tau} \text{ 对于无限多的整数对 } (q,r)\} \end{align*}$$ 是一个塞勒姆集。一个自然要问的问题是，当 $nd>1$ 和 $\tau>; d/n$。我们通过计算这些集合的傅立叶维数来证明答案是否定的。此外，我们证明了当 1美元。$nd=1$ 的情况，即非常近似的数字，仍未解决。