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

We study sets of bounded remainder for the two-dimensional continuous irrational rotation $(\{x_1+t\}, \{x_2+tα\})_{t \geq 0}$ in the unit square. In particular, we show that for almost all $α$ and every starting point $(x_1, x_2)$, every polygon $S$ with no edge of slope $α$ 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 $α$ and every starting point $(x_1, x_2)$. Finally we show that these assertions are, in some sense, best possible.

No reviews yet.

No comments yet.