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Sets of bounded discrepancy for multi-dimensional irrational rotation. (English) Zbl 1318.11097
Let \(\alpha=(\alpha_1, \ldots, \alpha_d)\) be a vector in \(\mathbb{R}^d\) and suppose that \(1, \alpha_1, \ldots, \alpha_d\) are linearly independent over the rationals. Let \(\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d\). For a Riemann-measurable set \(S\subset \mathbb{T}^d\) the discrepancy function of the sequence \(\{n\alpha\}\) is defined by \(D_n(S,x)=\sum_{k=0}^{n-1}\chi_S(x+k\alpha)-n \operatorname{mes} S\) for \(x\in \mathbb{T}^d\), where \(\chi_S\) is the indicator function of \(S\). A measurable set \(S\) is called a bounded remainder set (BRS) if there is a constant \(C=C(S,\alpha)\) such that \(|D_n(S,x)|\leq C\) for every \(n\) and almost every \(x\). The Hecke-Ostrowski-Kesten result states that in the case \(d=1\) an interval \(I\subset \mathbb{T}\) is a BRS if and only if its length belongs to \(\mathbb{Z}\alpha+\mathbb{Z}\). The authors prove that any parallelepiped in \(\mathbb{R}^d\) spanned by vectors belonging to \(\mathbb{Z}\alpha+\mathbb{Z}^d\) is a BRS (Theorem 1). Two measurable sets \(S\) and \(S'\) in \(\mathbb{R}^d\) are said to be equidecomposable if \(S\) can be partitioned into finitely many measurable subsets that can be reassembled by rigid motions to form a partition of \(S'\). The authors show that a Riemann measurable set \(S\) in \(\mathbb{R}^d\) is a BRS if and only if it is equidecomposable to some parallelepiped spanned by vectors in \(\mathbb{Z}\alpha+\mathbb{Z}^d\), using translation by vectors belonging to \(\mathbb{Z}\alpha+\mathbb{Z}^d\) (Corollary 3). Moreover in the case \(d=2\) they give a characterization of the convex polygons with bounded remainder.

MSC:
11K38 Irregularities of distribution, discrepancy
11J71 Distribution modulo one
52B45 Dissections and valuations (Hilbert’s third problem, etc.)
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