Pinned simplices and connections to product of sets on paraboloids
Journal
Indiana University Mathematics Journal
Journal Volume
74
Journal Issue
3
Start Page
647
End Page
668
ISSN
0022-2518
Date Issued
2025
Author(s)
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 Erdŏgan-Hart-Iosevich (2012), Greenleaf-Iosevich-Liu-Palsson (2015), and others.
Publisher
Indiana University Mathematics Journal
Type
journal article