On the self-similarity problem for Gaussian-Kronecker flows

It is shown that a countable symmetric multiplicative subgroup $G=-Hcup H$ with $Hsubsetmathbb{R}_+^ast$ 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 $s eqpm1$ 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.