Generalizations of Product-Free Subsets

Abstract

In this paper, we present some generalizations of Gowers’s result about product-free subsets of groups. 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 Epsilon A. Previous results showed that the size of any product-free subset of G is at most n/delta1/3, where delta 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 Epsilon A. We prove that the upper bound for the size of product-poor subsets is asymptotically the same as the size of product-free subsets. 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.