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，称子集 $S \subseteq G$ 为无乘积集，若方程 $xy=z$ 不存在 $x,y,z \in S$ 的解。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 $D$，则称群 $G$ 是 $D$ 拟随机的。Gowers 证明：在 $D$ 拟随机的有限群 $G$ 中，无乘积集的最大大小至多为 $|G|/D^{1/3}$，从而否定了 Babai 与 Sós 于 1985 年提出的一个长期猜想。对于特殊酉群 $G = SU(n)$，Gowers 指出其论证方法可推出：可测无乘积子集的测度上界为 $n^{-1/3}$。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：对于 $SU(n)$ 中测度至少为 $\exp(-cn^{1/3})$ 的可测子集，存在乘积混合性质；该乘积混合结果中的指数 $n^{1/3}$ 是紧致的。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。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 $D$ 拟随机的紧连通李群（其中 $D$ 至少为绝对常数），从而将无乘积集的结果推广至此类群。此外，我们展示了这些不等式在几何（即非阿贝尔 Brunn-Minkowski 型不等式）、混合时间以及紧致李群增长理论中的其他应用。