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The two-dimensional Hecke-Kesten problem. (Russian) Zbl 1306.11055
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 $r(\alpha,n,X)=\#\{i: 0\leq i\leq n, \{i\alpha\} \in X\}-n|X|$ 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.

##### MSC:
 11J71 Distribution modulo one 11H06 Lattices and convex bodies (number-theoretic aspects) 11K06 General theory of distribution modulo $$1$$
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