  Let $A$ be a compact set in $\mathbb{R}$, and $E=A^d\subset \mathbb{R}^d$. We
know from the Mattila-Sj\"olin's theorem if $\dim_H(A)>\frac{d+1}{2d}$, then
the distance set $\Delta(E)$ has non-empty interior. In this paper, we show
that the threshold $\frac{d+1}{2d}$ can be improved whenever $d\ge 5$.
