On a family of two-dimensional bounded remainder sets (Q2857878)

Let \(1, \alpha_1,\ldots,\alpha_m\) be linearly independent over \(\mathbb{Z},\) let \(\alpha=(\alpha_1,\ldots,\alpha_m)\). The set \(X\subset[0;1)^m\) is called a bounded remainder set if for NEWLINE\[ NEWLINE r(\alpha,n,X)=\#\{i: 1\leq i\leq n, \{i\alpha\} \in X\}-n|X|NEWLINE\] NEWLINE one has \(r(\alpha,n,X)=O(1).\) The case \(m=1\) is known as a Hecke-Kesten problem and has been already studied. For example a full description of bounded remainder intervals is known. However, the case \(m=2\) is much more complicated. Only few examples of bounded remainder sets are known but without estimates on \(r(\alpha,n,X)\). In the paper under review, for any vector \(\alpha\) an uncountable number of bounded remainder sets is constructed. Moreover, for such sets the estimates on \(r(\alpha,n,X)\) are obtained. The proof is based on a reformulation of the problem in terms of lattices.