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 nu
mber $s eqpm1$ 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.
We deal with Besicovitchs problem of existence of discrete orbits for transitive cylindrical transformations $T_varphi:(x,t)mapsto(x+alpha,t+varphi(x))$ where $Tx=x+alpha$ is an irrational rotation on the circle $T$ and $varphi:TtoR$ is continuous, i
.e. we try to estimate how big can be the set $D(alpha,varphi):={xinT:|varphi^{(n)}(x)|to+inftytext{as}|n|to+infty}$. We show that for almost every $alpha$ there exists $varphi$ such that the Hausdorff dimension of $D(alpha,varphi)$ is at least $1/2$. We also provide a Diophantine condition on $alpha$ that guarantees the existence of $varphi$ such that the dimension of $D(alpha,varphi)$ is positive. Finally, for some multidimensional rotations $T$ on $T^d$, $dgeq3$, we construct smooth $varphi$ so that the Hausdorff dimension of $D(alpha,varphi)$ is positive.
