  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.
