Bounded remainder sets on a two-dimensional torus (Q2857863)

Let \(1,\alpha_1,\ldots,\alpha_m\) be linearly independent over \(\mathbb{Z},\) let \(\alpha=(\alpha_1,\ldots,\alpha_m)\), \(x_0\in [0;1)^m\). The set \(X\subset [0;1)^m\) is called a bounded remainder set if for NEWLINE\[NEWLINE r(\alpha,n,X,x_0)=\#\{i: 0\leq i\leq n, \{i\alpha+x_0\} \in X\}-n|X|NEWLINE\]NEWLINE one has \(r(\alpha,n,X)=O(1)\). The case \(m=1\) is known as Hecke-Kesten problem and is very well investigated. For example, a full description of bounded remainder intervals is known. In the paper the case \(m=2\), \(x_0\neq 0\) is investigated. It is shown that this case can be treated in the same manner as the case \(m=2\), \(x_0=0\).