Bounded remainder sets for rotations on the adelic torus

In this paper we give an explicit construction of bounded remainder sets of all possible volumes, for any irrational rotation on the adelic torus $\mathbb A/\mathbb Q$. Our construction involves ideas from dynamical systems and harmonic analysis on the adeles, as well as a geometric argument which originated in the study of deformation properties of mathematical quasicrystals.Comment: 13 page

Projections of random covering sets

We show that, almost surely, the Hausdorff dimension $s_0$ of a random covering set is preserved under all orthogonal projections to linear subspaces with dimension $k>s_0$. The result holds for random covering sets with a generating sequence of ball-like sets, and is obtained by investigating orthogonal projections of a class of random Cantor sets.Comment: 17 page