{"title":"On the difference bases of $$\\mathbb {Z}_{m}$$","authors":"Yu Zhang","doi":"10.1007/s10998-024-00598-x","DOIUrl":null,"url":null,"abstract":"<p>For any positive integer <i>m</i>, let <span>\\(\\mathbb {Z}_m\\)</span> be the cyclic group of order <i>m</i>. For any subset <span>\\(A\\subseteq \\mathbb {Z}_{m}\\)</span> and any <span>\\(n\\in \\mathbb {Z}_{m}\\)</span>, let <span>\\(\\delta _{A}(n)=\\#\\{(a,b)|n=a-b, a\\in A, b\\in A\\}\\)</span>. In this paper, we prove that, for any positive integer <i>m</i>, there exists a subset <i>A</i> of <span>\\(\\mathbb {Z}_m\\)</span> such that <span>\\(\\delta _A (n)\\le 5\\)</span> for all <span>\\(n \\in \\mathbb {Z}_m\\)</span> with at most 3 exceptions, which improves a 2010 result of Y.–G. Chen & T. Sun.</p>","PeriodicalId":49706,"journal":{"name":"Periodica Mathematica Hungarica","volume":"44 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Periodica Mathematica Hungarica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10998-024-00598-x","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
For any positive integer m, let \(\mathbb {Z}_m\) be the cyclic group of order m. For any subset \(A\subseteq \mathbb {Z}_{m}\) and any \(n\in \mathbb {Z}_{m}\), let \(\delta _{A}(n)=\#\{(a,b)|n=a-b, a\in A, b\in A\}\). In this paper, we prove that, for any positive integer m, there exists a subset A of \(\mathbb {Z}_m\) such that \(\delta _A (n)\le 5\) for all \(n \in \mathbb {Z}_m\) with at most 3 exceptions, which improves a 2010 result of Y.–G. Chen & T. Sun.
期刊介绍:
Periodica Mathematica Hungarica is devoted to publishing research articles in all areas of pure and applied mathematics as well as theoretical computer science. To be published in the Periodica, a paper must be correct, new, and significant. Very strong submissions (upon the consent of the author) will be redirected to Acta Mathematica Hungarica.
Periodica Mathematica Hungarica is the journal of the Hungarian Mathematical Society (János Bolyai Mathematical Society). The main profile of the journal is in pure mathematics, being open to applied mathematical papers with significant mathematical content.