Product Mixing in Compact Lie Groups

TL;DR

The paper develops a robust analytic and representation-theoretic framework to bound the size of product-free sets in compact Lie groups, proving an exp(-c n^{1/3}) bound for SU(n) and a general exp(-c D(tilde G)^{1/3}) bound in large D-quasirandom compact connected Lie groups. Central to the approach are level-d inequalities and hypercontractivity, achieved via a novel Gaussian–Lie group coupling, Gram–Schmidt based noise operators, and a detailed degree-decomposition of L^2(G) into invariant subspaces V_{=d}. The work also introduces comfortable d-juntas to obtain explicit eigenfunction bases for low-degree representations, enabling sharp spectral control and product-mixing, Brunn–Minkowski-type inequalities, and diameter bounds for good and fine groups. By establishing that Sp(n), Spin(n), and SU(n) are good, and extending these techniques through grading and quotient structures, the authors provide new tools for growth, mixing, and equidistribution in a broad class of compact Lie groups, with potential applications in geometry and dynamics of homogeneous spaces.

Abstract

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 Sós 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.

Theorems & Definitions (188)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Conjecture 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10