Abstract
For any $\gamma\in(0,1)$ and any $\varepsilon>0$ we construct a cylindrical cascade over some irrational circle rotation with a $\gamma$-Holder function such that the Besicovitch condition holds and the Hausdorff dimension of the set of points in the circle having discrete orbits is more than $1-\gamma-\varepsilon$. This result gives the answers to some questions of K. Frączek and M. Lemanczyk [1].
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