A reverse Minkowski theorem

• 成果类型: Article
• 署名作者: Regev, Oded; Stephens-Davidowitz, Noah
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2024.199.1.1
• 发表日期: 2024
• 页码: 1-49
• 关键词: reduction theory; lattice; Ellipsoids; geometry; minima

摘要

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.