The two-dimensional Hecke-Kesten problem (Q2857842)

Let \(1,\alpha_1,\dots,\alpha_m\) be linearly independent over \(\mathbb{Z},\) let \(\alpha=(\alpha_1,\dots,\alpha_m).\) The set \(X\subset[0;1)^m\) is called a bounded remainder set if for NEWLINE\[NEWLINE r(\alpha,n,X)=\#\{i: 0\leq i\leq n, \{i\alpha\} \in X\}-n|X|NEWLINE\]NEWLINE one has \(r(\alpha,n,X)=O(1).\) The case \(m=1\) is known as Hecke-Kesten problem and is very well investigated. For example, a full description of bounded remainder intervals is known. But even 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 for any vector \(\alpha\) an uncountable number of bounded remainder sets is constructed. Moreover, for such sets estimates on \(r(\alpha,n,X)\) are obtained. The proof is based on a reformulation of the problem in terms of lattices. It is also proved that if \(X\) is a bounded remainder set for \(\alpha\) then \(X\) will be a bounded remainder set for infinitely many vectors \(\beta=(\alpha+b)/h\) where \(h\) is an integer and \(b\) belongs to the lattice.