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

TL;DR

This paper investigates sets of bounded remainder for continuous irrational rotations in the unit square, showing that most polygons and smooth convex sets are bounded remainder sets for almost all slopes and starting points.

Contribution

It establishes new results on bounded remainder sets for continuous irrational rotations, including polygons without edges of slope α and smooth convex sets, with optimality considerations.

Findings

Polygons with no edge of slope α are bounded remainder sets for almost all α.

Smooth convex sets with twice differentiable boundaries and positive curvature are bounded remainder sets.

The results are, in some sense, optimal.

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.