On the Mattila–Sjölin distance theorem for product sets

Abstract

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