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A reverse Minkowski theorem
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DOI：10.4007/annals.2024.199.1.1.png)
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Abstract
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We prove a conjecture due to Dadush, showing that if L subset of R-n is a lattice such that det (L ') >= 1 for all sublattices L 'subset of L, then & sum;(e-pi t2 parallel to y parallel to 2 <= 3/2, )(y is an element of L)where t:=10(logn+2). From this we derive bounds on the number of short lattice vectors, which can be viewed as a partial converse to Minkowski's celebrated first theorem. We also derive a bound on the covering radius.
Keywords:
lattices
geometry of numbers
Minkowski's theorem
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