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A reverse Minkowski theorem. (English) Zbl 07782629

This paper is dedicated to the proof of Dadush conjecture and its application to the Minkovski’s first theorem. Recall that the conjecture deals with lattices \(\mathcal L \subset \mathbb R^n\) whose all sublattices \(\mathcal L'\) satisfy \(\det \mathcal L' \ge 1\). It was conjectured by Dadush that such lattices satisfy \(\sum_{y\in \mathcal L} t^{-\pi t^2||y^2||}<3/2\) where \(t=1=\log n+2\). The authors derive bounds on the number of short lattice vectors, which can be viewed as a partial converse to Minkowski’s first theorem. Further they show a bound on the covering radius.

MSC:

52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
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