Corners over quasirandom groups

TL;DR

This paper proves that in large quasirandom groups, the proportion of corners within dense subsets approaches certainty as the group's quasirandomness increases, revealing a strong combinatorial structure.

Contribution

It establishes a new asymptotic result on the density of corners in dense subsets of quasirandom groups, extending combinatorial understanding in algebraic settings.

Findings

Density of corners approaches 1 as quasirandomness D increases

Large quasirandom groups contain many structured configurations

Results generalize classical combinatorial theorems to algebraic group contexts

Abstract

Let $G$ be a finite $D$-quasirandom group and $A \subset G^{k}$ a $\delta$-dense subset. Then the density of the set of side lengths $g$ of corners \[ \{(a_{1},\dots,a_{k}),(ga_{1},a_{2},\dots,a_{k}),\dots,(ga_{1},\dots,ga_{k})\} \subset A \] converges to $1$ as $D\to\infty$.