On the self-similarity problem for Gaussian-Kronecker flows

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.