On Hausdorff dimension of the set of closed orbits for a cylindrical transformation

We deal with Besicovitch’s problem of existence of discrete orbits for transitive cylindrical transformations $T_{\phi }:(x,t)\mapsto (x+\alpha ,t+\phi (x))$ where $Tx=x+\alpha$ is an irrational rotation on the circle $\text{T}$ and $\phi :\text{T}\to \mathbb{R}$ is continuous, i.e.\ we try to estimate how big can be the set $D(\alpha ,\phi ):=\{x\in \text{T}:\mathrm{\mid }{\phi }^{(n)}(x)\mathrm{\mid }\to +\mathrm{\infty }\text{as}\mathrm{\mid }n\mathrm{\mid }\to +\mathrm{\infty }\}$. We show that for almost every $\alpha$ there exists $\phi$ such that the Hausdorff dimension of $D(\alpha ,\phi )$ is at least $1\mathrm{/}2$. We also provide a Diophantine condition on $\alpha$ that guarantees the existence of $\phi$ such that the dimension of $D(\alpha ,\phi )$ is positive. Finally, for some multidimensional rotations $T$ on ${\text{T}}^d$, $d\ge 3$, we construct smooth $\phi$ so that the Hausdorff dimension of $D(\alpha ,\phi )$ is positive.