{"title":"On lattice extensions","authors":"Maxwell Forst, Lenny Fukshansky","doi":"10.1007/s00605-023-01935-x","DOIUrl":null,"url":null,"abstract":"<p>A lattice <span>\\(\\Lambda \\)</span> is said to be an extension of a sublattice <i>L</i> of smaller rank if <i>L</i> is equal to the intersection of <span>\\(\\Lambda \\)</span> with the subspace spanned by <i>L</i>. The goal of this paper is to initiate a systematic study of the geometry of lattice extensions. We start by proving the existence of a small-determinant extension of a given lattice, and then look at successive minima and covering radius. To this end, we investigate extensions (within an ambient lattice) preserving the successive minima of the given lattice, as well as extensions preserving the covering radius. We also exhibit some interesting arithmetic properties of deep holes of planar lattices.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-023-01935-x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
A lattice \(\Lambda \) is said to be an extension of a sublattice L of smaller rank if L is equal to the intersection of \(\Lambda \) with the subspace spanned by L. The goal of this paper is to initiate a systematic study of the geometry of lattice extensions. We start by proving the existence of a small-determinant extension of a given lattice, and then look at successive minima and covering radius. To this end, we investigate extensions (within an ambient lattice) preserving the successive minima of the given lattice, as well as extensions preserving the covering radius. We also exhibit some interesting arithmetic properties of deep holes of planar lattices.
如果 L 等于 \(\Lambda \)与 L 所跨子空间的交集,那么一个网格 \(\Lambda \)就可以说是一个秩较小的子网格 L 的扩展。我们首先证明给定网格的小确定性扩展的存在,然后研究连续最小值和覆盖半径。为此,我们研究了保留给定网格的连续最小值的扩展(在环境网格内),以及保留覆盖半径的扩展。我们还展示了平面网格深洞的一些有趣的算术性质。