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BR-sets. (Russian. English summary) Zbl 1436.11095
Let \(\alpha_1,\ldots,\alpha_d\) be irrational numbers and suppose that \(1,\alpha_1,\ldots,\alpha_d\) are linearly independent over \(\mathbb{Z}\). The set \(X\subset [0;1)^d\) is called a bounded remainder set if the quantity \[ \delta(i,x_0)=\sharp\{ j:0\leq j<i:(\{j\alpha_1+x_0\},\ldots,\{j\alpha_d+x_0\})\in X\}-ivol(X) \] is bounded by an absolute constant.
The paper is an expository of previous author’s results on bounded remainder sets for \(d=2\) and \(d=3\), mainly devoted to constructing new examples of bounded remainder sets and finding exact bounds for \(\delta(i,x_0)\) for these sets.

MSC:
11K38 Irregularities of distribution, discrepancy
11J71 Distribution modulo one
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