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$G$为一个群，称子集$S\subseteq G$为无乘积集，若方程$xy=z$在$S$中无解$x,y,z$。对于$D\in\mathbb{N}$，若群$G$的所有非平凡复不可约表示的最小维数至少为$D$，则称$G$为$D$-拟随机群。高尔斯证明：在$D$-拟随机有限群$G$中，无乘积集的最大尺寸至多为$|G|/D^{1/3}$。这一结果否定了鲍包伊与沙什自1985年提出的一个长期猜想。对于特殊酉群$G=SU(n)$，高尔斯指出其论证可得可测无乘积集测度的上界为$n^{-1/3}$。本文将该上界改进为$\exp(-cn^{1/3})$，其中$c>0$为绝对常数。事实上，我们建立了更强的结论：$SU(n)$中测度至少为$\exp(-cn^{1/3})$的可测子集满足乘积混合性质，且该指数中的$n^{1/3}$是最优的。我们的方法引入新型超压缩不等式，该不等式表明小集指示函数的非阿贝尔傅里叶谱集中于高维不可约表示。这些超压缩不等式通过表示论、调和分析、随机矩阵论与微分几何方法获得。我们将超压缩不等式从$SU(n)$推广至任意$D$不小于绝对常数的$D$-拟随机紧连通李群，从而将无乘积集的相关结论推广至该类群。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其他应用。