Generalizations of product-free subsets

TL;DR

This paper extends bounds on the size of product-free subsets in groups to more general 'product-poor' subsets, providing tighter bounds and exploring multiple subset interactions.

Contribution

It generalizes Gowers' upper bound to product-poor subsets and introduces a framework for multiple subsets with various product constraints.

Findings

Upper bounds match known lower bounds in many group families

Generalization to multiple subsets with different product constraints

Extension of product-free concept to broader subset interactions

Abstract

For any group G of order n, a subset A of G is said to be product-free if there is no solution of the equation ab=c with a,b,c in A. Previous results of Gowers showed that the size of any product-free subset of G is at most n/d^(1/3), where d is the smallest dimension of a nontrivial representation of G. However, this upper bound does not match the best lower bound. We will generalize the upper bound to the case of product-poor subsets A, in which the equation ab=c is allowed to have a few solutions with a,b,c in A. We prove that the upper bound for the size of product-poor subsets matches the best lower bound in many families of groups. We will also generalize the concept of product-free to the case in which we have many subsets of a group, and different constraints about products of the elements in the subsets.