A Mattila–Sjölin theorem for simplices in low dimensions

In this paper we show that if a compact set E ⊂ Rᵈ E ⊂ R d , d ≥ 3 d ≥ 3 , has Hausdorff dimension greater than (4k-1)/4kd+1/4 ( 4 k - 1 ) 4 k d + 1 4 when 3 ≤ d<k(k+3)/(k-1) 3 ≤ d < k ( k + 3 ) ( k - 1 ) or d- 1/k-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 Δ ₖ(E) = \ t = ( tᵢⱼ ) : |xᵢ-xⱼ|=tᵢⱼ; \ xᵢ,xⱼ ∈ E; \ 0≤ i < j ≤ k \ ⊂ Rk(k+1)/2 aligned 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 <d 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.