Product mixing in the alternating group

TL;DR

This paper establishes a product-mixing theorem for the alternating group, providing bounds on the size of product-free subsets and advancing understanding of group structure using Fourier analysis and measure concentration.

Contribution

It proves a new one-sided product-mixing theorem for the alternating group, improving bounds on product-free subset densities and applying advanced harmonic analysis tools.

Findings

At least (1+o(1))αβγ|A_n|^2 solutions to xy=z under certain density conditions

Largest product-free subset of A_n has density at most n^{-1/2}( ext{log} n)^{7/2}

Improves previous bounds from n^{-1/3} to near optimal up to logarithmic factors

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.