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

Sigrid Grepstad¹ ; Gerhard Larcher¹

¹ Institute of Financial Mathematics and Applied Number Theory Johannes Kepler University Linz Altenbergerstr. 69 A-4040 Linz, Austria

Acta Arithmetica, Tome 176 (2016) no. 4, pp. 365-395

Résumé

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.

Détail

DOI : 10.4064/aa8453-8-2016

Keywords: study sets bounded remainder two dimensional continuous irrational rotation alpha geq unit square particular almost alpha every starting point every polygon edge slope alpha set bounded remainder moreover every convex set whose boundary twice continuously differentiable positive curvature every point bounded remainder set almost alpha every starting point finally these assertions sense best possible