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（product-free），若方程$xy=z$在$S$中无解，即不存在$x,y,z\in S$满足该方程。对$D\in\mathbb{N}$，称群$G$是$D$-拟随机的（$D$-quasirandom），若其所有非平凡复不可约表示的最小维数至少为$D$。Gowers证明，在有限$D$-拟随机群$G$中，无乘积集的最大规模至多为$|G|/D^{1/3}$。这一结果否定了Babai与Sós自1985年起长期存在的猜想。对于特殊酉群$G=SU(n)$，Gowers指出其论证可推出可测无乘积子集的上界为$n^{-1/3}$（按Haar测度）。本文将其上界改进至$\exp(-cn^{1/3})$，其中$c>0$为绝对常数。事实上，我们建立了更强的结论：对$SU(n)$中测度至少为$\exp(-cn^{1/3})$的可测子集，其满足乘积混合性质（product-mixing）；该乘积混合结果中指数上的$n^{1/3}$是最优的。我们的方法涉及引入新型超收缩不等式（hypercontractive inequality），该不等式表明小集合示性函数的非交换Fourier谱集中于高维不可约表示。这些超收缩不等式通过表示论、调和分析、随机矩阵理论与微分几何中的方法得到。我们将超收缩不等式从$SU(n)$推广到任意$D$不小于某个绝对常数的$D$-拟随机紧连通李群，从而将关于无乘积集的结论推广至此类群。此外，我们还展示了这些不等式在几何（即非交换Brunn-Minkowski型不等式）、混合时间及紧李群增长理论中的若干其他应用。