Abstract
The aim of this paper is to generalize Minkowski’s theorem. This theorem is usually stated for a centrally symmetric convex body and a lattice both included in \(\mathbf {R}^n\). In some situations, one may replace the lattice by a more general set for which a notion of density exists. In this paper, we prove a Minkowski theorem for quasicrystals, which bounds from below the frequency of differences appearing in the quasicrystal and belonging to a centrally symmetric convex body. The last part of the paper is devoted to quite natural applications of this theorem to Diophantine approximation and to discretization of linear maps.
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Notes
Here, the mean is taken with respect to Lebesgue measure instead of counting measure as in Definition 2.2.
That is, \(\pi (x)\) is (one of the) point(s) of \(\mathbf {Z}^n\) the closest from x for the Euclidean norm.
A property concerning elements of a topological set X is called generic if it is satisfied on at least a countable intersection of open and dense sets. In particular, Baire theorem implies that if this space is complete (as here), then this property is true on a dense subset of X.
The set of sequences of matrices is endowed with the norm \(\Vert (P_k)_{k\ge 1}\Vert = \sup _{k\ge 1} \Vert P_k\Vert \), making it a complete space (see Note 3).
Unfortunately, there is a misprint in the first inequality of (ii): at the left side a constant depending only on \(\Vert P^{-1}\Vert \) is missing.
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Acknowledgements
The first author was funded by an IMPA/CAPES grant. The second author is funded by the French Agence Nationale de la Recherche (ANR), under Grant ANR-13-BS01-0005 (Project SPADRO). The authors warmly thank Samuel Petite, and the anonymous referee for their numerous thoughtful comments.
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Guihéneuf, PA., Joly, É. A Minkowski Theorem for Quasicrystals. Discrete Comput Geom 58, 596–613 (2017). https://doi.org/10.1007/s00454-017-9864-0
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DOI: https://doi.org/10.1007/s00454-017-9864-0