Pinned simplices and connections to product of sets on paraboloids

TL;DR

This paper improves the understanding of how the size of sets on paraboloids influences the diversity of geometric configurations they determine, leading to better thresholds for the existence of various simplices.

Contribution

It provides new, sharper dimensional thresholds for the occurrence of dot product sets and congruence classes of simplices within compact subsets of Euclidean space.

Findings

Enhanced thresholds for dot product sets on paraboloids.

Improved conditions for sets to determine many congruence classes of simplices.

Advances over previous results by Erdogan-Hart-Iosevich and Greenleaf-Iosevich-Liu-Palsson.

Abstract

In this paper we obtain improved dimensional thresholds for dot product sets corresponding to compact subsets of a paraboloid. As a direct application of these estimates, we obtain significant improvements to the best known dimensional thresholds that guarantee that a given compact subset of Euclidean space determines a positive proportion of all possible congruence classes of simplexes. In many regimes this improves the results previously obtained by Erdogan-Hart-Iosevich (\cite{EHI}), Greenleaf-Iosevich-Liu-Palsson (\cite{GILP}) and others.