Sets of bounded remainder for a continuous irrational rotation on $[0,1]^2$
Volume 176 / 2016
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.
