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Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices. (English) Zbl 1341.11043
Given a cut-and-project setup corresponding to an \(\mathbb{R}^{d}\) action on \(\mathbb{R}^{k}\), the authors construct an infinite collection of \((k-d)\)-dimensional “windows” which each yield a cut-and-project set with bounded distance from a lattice. Furthermore, for a given minimal torus-rotation they explicitly construct an infinite collection of regions in the \(s\)-dimensional tours, for which the corresponding discrepancies over each of the regions achieve the minimal possible asymptotic bound. The proof provides a new construction (based on a condition due to G. Rauzy [Sémin. Théor. Nombres, Univ. Bordeaux I 1983–1984, Exp. No. 24, 12 p. (1984; Zbl 0547.10044)]) of an infinite family of bounded remainder sets for any irrational rotation on the \(s\)-dimensional torus. For recent progress on bonded remainder sets for irrational tonal relations see also [S. Grepstad and N. Lev, Geom. Funct. Anal. 25, No. 1, 87–133 (2015; Zbl 1318.11097)].

MSC:
11K06 General theory of distribution modulo \(1\)
11K38 Irregularities of distribution, discrepancy
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