Weak and strong regularity, compactness, and approximation of polynomials

Abstract

Let $X$ be an inner product space, let $G$ be a group of orthogonal transformations of $X$, and let $R$ be a bounded $G$-stable subset of $X$. We define very weak and very strong regularity for such pairs $(R,G)$ (in the sense of Szemer\'edi's regularity lemma), and prove that these two properties are equivalent. Moreover, these properties are equivalent to the compactness of the space $(B(H),d_R)/G$. Here $H$ is the completion of $X$ (a Hilbert space), $B(H)$ is the unit ball in $H$, $d_R$ is the metric on $H$ given by $d_R(x,y):=\sup_{r\in R}|<r,x-y>|$, and $(B(H),d_R)/G$ is the orbit space of $(B(H),d_R)$ (the quotient topological space with the $G$-orbits as quotient classes). As applications we give Szemer\'edi's regularity lemma, a related regularity lemma for partitions into intervals, and a low rank approximation theorem for homogeneous polynomials.