Weak and strong regularity, compactness, and approximation of polynomials

TL;DR

This paper establishes the equivalence of weak and strong regularity conditions for group actions on inner product spaces, linking these to compactness and applying these results to regularity lemmas and polynomial approximation.

Contribution

It introduces new regularity concepts for pairs (R,G), proves their equivalence, and connects them to compactness and polynomial approximation in Hilbert spaces.

Findings

Weak and strong regularity are equivalent for (R,G) pairs.

Regularity properties are equivalent to the compactness of a certain orbit space.

Applications include Szemerédi's regularity lemma and low rank polynomial approximation.

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.