{"title":"Lonely Points in Simplices.","authors":"Maximilian Jaroschek, Manuel Kauers, Laura Kovács","doi":"10.1007/s00454-022-00428-2","DOIUrl":null,"url":null,"abstract":"<p><p>Given a lattice <math><mrow><mi>L</mi> <mo>⊆</mo> <msup><mi>Z</mi> <mi>m</mi></msup> </mrow> </math> and a subset <math><mrow><mi>A</mi> <mo>⊆</mo> <msup><mi>R</mi> <mi>m</mi></msup> </mrow> </math> , we say that a point in <i>A</i> is <i>lonely</i> if it is not equivalent modulo <math><mi>L</mi></math> to another point of <i>A</i>. We are interested in identifying lonely points for specific choices of <math><mi>L</mi></math> when <i>A</i> is a dilated standard simplex, and in conditions on <math><mi>L</mi></math> which ensure that the number of lonely points is unbounded as the simplex dilation goes to infinity.</p>","PeriodicalId":50574,"journal":{"name":"Discrete & Computational Geometry","volume":"69 1","pages":"4-25"},"PeriodicalIF":0.6000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9805990/pdf/","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete & Computational Geometry","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00454-022-00428-2","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2022/9/29 0:00:00","PubModel":"Epub","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
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Abstract
Given a lattice and a subset , we say that a point in A is lonely if it is not equivalent modulo to another point of A. We are interested in identifying lonely points for specific choices of when A is a dilated standard simplex, and in conditions on which ensure that the number of lonely points is unbounded as the simplex dilation goes to infinity.
期刊介绍：
Discrete & Computational Geometry (DCG) is an international journal of mathematics and computer science, covering a broad range of topics in which geometry plays a fundamental role. It publishes papers on such topics as configurations and arrangements, spatial subdivision, packing, covering, and tiling, geometric complexity, polytopes, point location, geometric probability, geometric range searching, combinatorial and computational topology, probabilistic techniques in computational geometry, geometric graphs, geometry of numbers, and motion planning.
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