On the self-similarity problem for Gaussian-Kronecker flows

TL;DR

This paper characterizes the groups of self-similarities for Gaussian-Kronecker flows, showing they are linked to additive independence over rationals and that transcendental numbers are precisely the scales of self-similarity.

Contribution

It provides a complete characterization of self-similarity groups for Gaussian-Kronecker flows and constructs flows with prescribed self-similarity groups.

Findings

A countable symmetric subgroup is the self-similarity group iff its positive part is additively $ ext{Q}$-independent.

A real number is a scale of self-similarity iff it is transcendental.

Any countable symmetric subgroup can be realized as a self-similarity group of a Gaussian flow with simple spectrum.

Abstract

It is shown that a countable symmetric multiplicative subgroup $G=-H\cup H$ with $H\subset\mathbb{R}_+^\ast$ is the group of self-similarities of a Gaussian-Kronecker flow if and only if $H$ is additively $\mathbb{Q}$-independent. In particular, a real number $s\neq\pm1$ is a scale of self-similarity of a Gaussian-Kronecker flow if and only if $s$ is transcendental. We also show that each countable symmetric subgroup of $\mathbb{R}^\ast$ can be realized as the group of self-similarities of a simple spectrum Gaussian flow having the Foias-Stratila property.