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Abstract: For any group G of order n, a subset A of G is said to be product-free ifthere is no solution of the equation ab=c with a,b,c in A. Previous results ofGowers showed that the size of any product-free subset of G is at mostn-d^1-3, where d is the smallest dimension of a nontrivial representation ofG. However, this upper bound does not match the best lower bound. We willgeneralize the upper bound to the case of product-poor subsets A, in which theequation ab=c is allowed to have a few solutions with a,b,c in A. We prove thatthe upper bound for the size of product-poor subsets matches the best lowerbound in many families of groups. We will also generalize the concept ofproduct-free to the case in which we have many subsets of a group, anddifferent constraints about products of the elements in the subsets.

Author: Kiran S. Kedlaya, Xuancheng Shao


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