In this paper we show that if a compact set , , has Hausdorff dimension greater than when or when , then the set of congruence class of simplices with vertices in has nonempty interior. By set of congruence class of simplices with vertices in we mean where . This result improves the previous best results 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 has nonempty interior when as well as extending to all simplices. The present work can be thought of as an extension of the Mattila-Sjölin theorem which establishes a non-empty interior for the distance set instead of the set of congruence classes of simplices.
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