  We deal with Besicovitch's 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 $\varphi:\T\to\R$ is continuous,
i.e.\ we try to estimate how big can be the set
$D(\alpha,\varphi):=\{x\in\T:|\varphi^{(n)}(x)|\to+\infty\text{as}|n|\to+\infty\}$.
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 $\T^d$, $d\geq3$, we construct smooth
$\varphi$ so that the Hausdorff dimension of $D(\alpha,\varphi)$ is positive.
