Bounded remainder sets for rotations on higher-dimensional adelic tori

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 $𝔸^{d} ∕ ℚ^{d}$ . 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 $ℝ^{d} ∕ ℚ^{d}$ .

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Keywords

bounded remainder sets, Birkhoff sums, adeles

Mathematical Subject Classification

Primary: 11J61, 11K38, 37A44

Milestones

Received: 25 September 2020

Accepted: 8 January 2021

Published: 23 June 2021

Authors

Akshat Das
Department of Mathematics University of Houston Houston, TX United States

Joanna Furno
Mathematical Sciences DePaul University Chicago, IL United States

Alan Haynes
Department of Mathematics University of Houston Houston, TX United States

Vol. 10, No. 2, 2021

Akshat Das, Joanna Furno and Alan Haynes

Abstract