Solving xz=yy in certain subsets of finite groups

TL;DR

This paper establishes an upper bound on the size of subsets within finite groups that contain no non-trivial solutions to the equation xz=yy, demonstrating a logarithmic decay related to the group's size.

Contribution

It provides a new bound on the size of solution-free subsets in finite groups for the equation xz=yy, advancing understanding of algebraic structures in combinatorics.

Findings

Subsets with no solutions are significantly smaller than the group itself.

The bound involves a logarithmic factor, specifically |G|/(log log |G|)^c.

The result applies to certain subsets of finite groups, not necessarily all.

Abstract

We show that if G is a finite group and A is a subset of G with no non-trivial solutions to xz=yy then |A| < |G|/(log log |G|)^c.