A Mattila–Sjölin theorem for simplices in low dimensions

Abstract In this paper we show that if a compact set E {R}ᵈ E ⊂ R d, d 3 d ≥ 3, has Hausdorff dimension greater than (4k-1) 4kd+14 (4 k - 1) 4 k d + 1 4 when 3 d 3 ≤ d k (k + 3) (k - 1) or d- 1k-1 d - 1 k - 1 when k (k+3) (k-1) d k (k + 3) (k - 1) ≤ 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 aligned ₊ (E) = \{ t = (t₈₉): |xᵢ-xⱼ|=t₈₉; \ xᵢ, xⱼ E; \ 0 i Δ k (E) = t = t ij: | x i - x j | = t ij ; x i, x j ∈ E ; 0 ≤ i j ≤ k ⊂ R k (k + 1) 2 where 2 k 2 ≤ k d. 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 E has nonempty interior when d=3 d = 3 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.