In this paper we show that if a compact set E⊂Rd, d≥3, has Hausdorff dimension greater than 4k(4k−1)​d+41​ when
3≤d<(k−1)k(k+3)​ or d−k−11​ when
(k−1)k(k+3)​≤d, then the set of congruence class of simplices
with vertices in E has nonempty interior. By set of congruence class of
simplices with vertices in E we mean Δk​(E)={t=(tij​):∣xi​−xj​∣=tij​; xi​,xj​∈E; 0≤i<j≤k}⊂R2k(k+1)​ where 2≤k<d. This result
improves our previous work in the sense that we now can obtain a Hausdorff
dimension threshold which allow us to guarantee that the set of congruence
class of triangles formed by triples of points of E has nonempty interior
when d=3 as well as extending to all simplices. The present work can be
thought of as an extension of the Mattila-Sj\"olin theorem which establishes a
non-empty interior for the distance set instead of the set of congruence
classes of simplices.Comment: 20 pages, 3 figure