Generalizations of Product-Free Subsets
Name
Kedlaya Generalization of Product-free....pdf
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Author(s)
Kedlaya, Kiran S.
Shao, Xuancheng
Date Issued
January 2009
Journal
Contemporary Mathematics (In Communicating Mathematics)
Publisher
American Mathematical Society
Citation
Kedlaya, Kiran S. and Xuancheng Shao. "Generalizations of Product-Free Subsets" in Communicating Mathematics: a conference in honor of Joseph A. Gallian's 65th birthday, July 16-19, 2007, University of Minnesota, Duluth, Minnesota. Timothy Y. Chow, Daniel C. Isaksen, editors. Providence, R.I.: American Mathematical Society, c2009, 338 pp. (Contemporary Mathematics ; v.479)
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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http://www.jointmathematicsmeetings.org/bookstore?fn=20&arg1=conmseries&ikey=CONM-479
