A reverse Minkowski theorem | Annals of Mathematics

IF 8.3 2区材料科学 Q1 MATERIALS SCIENCE, MULTIDISCIPLINARY

ACS Applied Materials & Interfaces Pub Date : 2023-12-29 DOI:10.4007/annals.2024.199.1.1

Oded Regev, Noah Stephens-Davidowitz

引用次数: 0

Abstract

We prove a conjecture due to Dadush, showing that if $\mathcal{L} \subset \mathbb{R}^n$ is a lattice such that $\mathrm{det}(\mathcal{L}’)\ge 1$ for all sublattices $\mathcal{L}’ \subseteq \mathcal{L}$, then $\sum_{\mathbf{y}\in \mathcal{L}} e^{-\pi t^2 \|\mathbf{y} \|^2} \le 3/2$, where $t := 10(\log n + 2)$. 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.

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反向闵科夫斯基定理 | 数学年鉴

我们证明了由达杜什提出的一个猜想，即如果 $\mathcal{L}\子集 $\mathbb{R}^n$ 是一个网格，使得 $\mathrm{det}(\mathcal{L}')\ge 1$ 对于所有子网格 $\mathcal{L}' \subseteq \mathcal{L}$，那么 $\sum_{\mathbf{y}\in \mathcal{L}} e^{-\pi t^2 \|\mathbf{y}|^2}。\|^2}\le 3/2$, 其中 $t := 10(\log n + 2)$.由此我们推导出短网格向量数的边界，这可以看作是闵科夫斯基著名的第一定理的部分逆定理。我们还推导出了覆盖半径的约束。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

ACS Applied Materials & Interfaces 工程技术-材料科学：综合

CiteScore

16.00

自引率

6.30%

发文量

4978

审稿时长

1.8 months

期刊介绍： ACS Applied Materials & Interfaces is a leading interdisciplinary journal that brings together chemists, engineers, physicists, and biologists to explore the development and utilization of newly-discovered materials and interfacial processes for specific applications. Our journal has experienced remarkable growth since its establishment in 2009, both in terms of the number of articles published and the impact of the research showcased. We are proud to foster a truly global community, with the majority of published articles originating from outside the United States, reflecting the rapid growth of applied research worldwide.