On polynomial $d$-chaos via $d$-dissociated character subsystems on compact abelian groups

TL;DR

This paper investigates polynomial $d$-chaoses on compact abelian groups, proving they form $q$-lacunary and Sidon systems, with implications for harmonic analysis and functional systems.

Contribution

It establishes that polynomial $d$-chaoses from $d$-dissociated character subsystems are $q$-lacunary and Sidon, advancing understanding of their structure.

Findings

Polynomial $d$-chaoses are $q$-lacunary systems.

They are also $2d/(d+1)$-Sidon systems.

Results apply to $d$-dissociated subsystems on compact abelian groups.

Abstract

In this paper, we study polynomial chaoses of degree $d$ constructed from sequences of functions; that is, sets of all possible $d$-fold products of sequence elements, allowing repeated factors. The tetrahedral chaos of degree $d$ is defined as the subset consisting of products with pairwise distinct factors. We prove that polynomial $d$-chaoses (and, consequently, the tetrahedral chaoses) with respect to $d$-dissociated subsystems of characters on compact abelian groups are $q$-lacunary and $2d/(d+1)$-Sidon systems.