A Mattila-Sjölin theorem for simplices in low dimensions

2022/08/12 by Eyvindur A. Palsson, Palsson, Eyvindur Ari, Francisco Romero Acosta +1 · 1 citation
Mathematics · Engineering · Computer Science · #Limits and Structures in Graph Theory #graph theory and CDMA systems #Advanced Graph Theory Research

paper · pdf · doi:10.48550/arxiv.2208.07198

Abstract

In this paper we show that if a compact set E ⊂ ℝ^d, d ≥ 3, has Hausdorff dimension greater than ((4k-1))/(4k)d+(1)/(4) when 3 ≤ d<(k(k+3))/((k-1)) or d- (1)/(k-1) when (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 Δ_k(E) = \ t = (t_ij) : |x_i-x_j|=t_ij ; x_i,x_j ∈ E ; 0≤ i lt; j ≤ k \ ⊂ ℝ^(k(k+1))/(2) where 2 ≤ k

Cited by

• Distribution of simplices in the discrete and continuous settings

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