Product mixing in the alternating group

Abstract

We prove the following one-sided product-mixing theorem for the alternating group: Given subsets $X,Y,Z\subset A_n$ of densities $\alpha ,\beta ,\gamma$ satisfying $\min (\alpha \beta ,\alpha \gamma ,\beta \gamma )\gg n^{-1}(\log n{)}^7$, there are at least $(1+o(1))\alpha \beta \gamma |A_n{|}^2$ solutions to $xy=z$ with $x\in X,y\in Y,z\in Z$. One consequence is that the largest product-free subset of $A_n$ has density at most $n^{-1/2}(\log n{)}^{7/2}$, which is best possible up to logarithms and improves the best previous bound of $n^{-1/3}$ due to Gowers. The main tools are a Fourier-analytic reduction noted by Ellis and Green to a problem just about the standard representation, a Brascamp--Lieb-type inequality for the symmetric group due to Carlen, Lieb, and Loss, and a concentration of measure result for rearrangements of inner products.

19 pages. Reformatted for Discrete Analysis but otherwise identical to the previous version

References in corpus (1)

• A Sharp analog of Young's Inequality on $S^N$ and Related Entropy Inequalities