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.
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