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

A finite subset of a Euclidean space is called an s-distance set if there exist exactly s values the distances between two distinct points in set. In this paper, we prove that maximum cardinality among all 5-distance sets $$\mathbb {R}^3$$ 20, and every with 20 similar to vertex regular dodecahedron.