Bounded remainder sets for rotations on higher dimensional adelic tori

In this paper we give a simple, explicit construction of polytopal bounded remainder sets of all possible volumes, for any irrational rotation on the $d$ dimensional adelic torus ${\mathbb{A}}^d\mathrm{/}{\mathbb{Q}}^d$. Our construction involves ideas from dynamical systems and harmonic analysis on the adeles, as well as a geometric argument that reduces the existence argument to the case of an irrational rotation on the torus ${\mathbb{R}}^d\mathrm{/}{\mathbb{Q}}^d$.