# '22

36th Summer Topology Conference

July 18-22, 2022

University of Vienna, Department of Mathematics
Oskar-Morgenstern-Platz 1, 1090 Vienna, AUSTRIA

### "Bounded remainder sets for rotations on compact groups"

#### Das, Akshat

Bounded remainder sets for a dynamical system are sets for which the Birkhoff averages of return times differ from the expected values by at most a constant amount. These sets are rare and important objects which have been studied, especially in the context of Diophantine approximation, for over 100 years. In the last few years, there have been a number of results which have culminated in explicit constructions of bounded remainder sets for toral rotations in any dimension, of all possible allowable volumes. In this talk, we are going to give a survey of these results, the recent constructions of bounded remainder sets for rotations on the adelic torus by Alan Haynes, Joanna Furno and Henna Koivusalo and finally give a brief description of a simple, explicit construction of polytopal bounded remainder sets of all possible volumes, for any irrational rotation on the $d$ dimensional adelic torus $\mathbb{A}^d/\mathbb{Q}^d$. 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 $\mathbb{R}^d/\mathbb{Q}^d$. This is joint work with Joanna Furno and Alan Haynes.

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