If $G$ is a group, we say a subset $S$ of $G$ is product-free if the equation $xy=z$ has no solutions with $x,y,z \in S$. For $D \in \mathbb{N}$, a group $G$ is said to be $D$-quasirandom if the minimal dimension of a nontrivial complex irreducible representation of $G$ is at least $D$. Gowers showed that in a $D$-quasirandom finite group $G$, the maximal size of a product-free set is at most $|G|/D^{1/3}$. This disproved a longstanding conjecture of Babai and S\'os from 1985. For the special unitary group, $G=SU(n)$, Gowers observed that his argument yields an upper bound of $n^{-1/3}$ on the measure of a measurable product-free subset. In this paper, we improve Gowers' upper bound to $\exp(-cn^{1/3})$, where $c>0$ is an absolute constant. In fact, we establish something stronger, namely, product-mixing for measurable subsets of $SU(n)$ with measure at least $\exp(-cn^{1/3})$; for this product-mixing result, the $n^{1/3}$ in the exponent is sharp. Our approach involves introducing novel hypercontractive inequalities, which imply that the non-Abelian Fourier spectrum of the indicator function of a small set concentrates on high-dimensional irreducible representations. Our hypercontractive inequalities are obtained via methods from representation theory, harmonic analysis, random matrix theory and differential geometry. We generalize our hypercontractive inequalities from $SU(n)$ to an arbitrary $D$-quasirandom compact connected Lie group for $D$ at least an absolute constant, thereby extending our results on product-free sets to such groups. We also demonstrate various other applications of our inequalities to geometry (viz., non-Abelian Brunn-Minkowski type inequalities), mixing times, and the theory of growth in compact Lie groups.

翻译：设$G$是一个群，称其子集$S$为无乘积集，若方程$xy=z$不存在解$x,y,z \in S$。对于$D \in \mathbb{N}$，若$G$的非平凡复不可约表示的最小维数至少为$D$，则称$G$是$D$-拟随机的。高尔斯证明，在$D$-拟随机有限群$G$中，最大无乘积集的元素个数不超过$|G|/D^{1/3}$，这推翻了巴拜与绍什自1985年以来的一个长期猜想。对于特殊酉群$G=SU(n)$，高尔斯指出其论证可给出可测无乘积集测度的上界$n^{-1/3}$。本文将该上界改进为$\exp(-cn^{1/3})$，其中$c>0$为绝对常数。实际上，我们建立了更强的结果：对于$SU(n)$中测度至少为$\exp(-cn^{1/3})$的可测子集，乘积混合性质成立，且此乘积混合结果中指数$n^{1/3}$是最优的。我们的方法涉及引入新型超压缩不等式，该不等式表明小集合示性函数的非阿贝尔傅里叶谱集中于高维不可约表示上。这些超压缩不等式通过表示论、调和分析、随机矩阵理论与微分几何的方法得到。我们将超压缩不等式从$SU(n)$推广到任意$D$不小于某绝对常数的$D$-拟随机紧连通李群，从而将关于无乘积集的结果推广至这类群。此外，我们还展示了这些不等式在几何(即非阿贝尔Brunn-Minkowski型不等式)、混合时间及紧李群增长理论中的多种其他应用。