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Sets of bounded remainder for a continuous irrational rotation on $[0,1]^2$

Volume 176 / 2016

Sigrid Grepstad, Gerhard Larcher Acta Arithmetica 176 (2016), 365-395 MSC: 11K38, 11J71. DOI: 10.4064/aa8453-8-2016 Published online: 26 October 2016

Abstract

We study sets of bounded remainder for the two-dimensional continuous irrational rotation $(\{x_1+t\}, \{x_2+t\alpha \})_{t \geq 0}$ in the unit square. In particular, we show that for almost all $\alpha$ and every starting point $(x_1, x_2)$, every polygon $S$ with no edge of slope $\alpha$ is a set of bounded remainder. Moreover, every convex set $S$ whose boundary is twice continuously differentiable with positive curvature at every point is a bounded remainder set for almost all $\alpha$ and every starting point $(x_1, x_2)$. Finally we show that these assertions are, in some sense, best possible.

Authors

  • Sigrid GrepstadInstitute of Financial Mathematics and Applied Number Theory
    Johannes Kepler University Linz
    Altenbergerstr. 69
    A-4040 Linz, Austria
    e-mail
  • Gerhard LarcherInstitute of Financial Mathematics and Applied Number Theory
    Johannes Kepler University Linz
    Altenbergerstr. 69
    A-4040 Linz, Austria
    e-mail

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