An elementary approach to simplexes in thin subsets of Euclidean space

2016/08/16 by Greenleaf, Allan, Iosevich, Alex, Liu, Bochen +1 · 1 citation
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1608.04777

Abstract

We prove that if the Hausdorff dimension of E ⊂ \Bbb R^d, d ≥ 3, is greater than min \ (dk+1)/(k+1), (d+k)/(2) \, then the k+1 \choose 2-dimensional Lebesgue measure of T_k(E), the set of congruence classes of k-dimensional simplexes with vertices in E, is positive. This improves the best bounds previously known, decreasing the (d+k+1)/(2) threshold obtained in Erdoğan-Hart-Iosevich (2012) to (d+k)/(2) via a different and conceptually simpler method. We also give a simpler proof of the d-(d-1)/(2d) threshold for d-dimensional simplexes obtained in Greenleaf-Iosevich (2012), Grafakos-Greenleaf-Iosevich-Palsson (2015).

Cited by

• Distribution of simplices in the discrete and continuous settings

Related

• Pinned simplices and connections to product of sets on paraboloids
• On thin carpets for doubling measures
• A Mattila-Sjölin theorem for simplices in low dimensions
• An Elementary Proof of the 3 Dimensional Simplex Mean Width Conjecture
• On volumes determined by subsets of Euclidean space