{"title":"Quasi-periodicity of \\(\\mathbb {Z}_{p^an_0}\\)","authors":"W. Zhou","doi":"10.1007/s10474-023-01361-3","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>p</i><sup><i>a</i></sup> be a prime power and <i>n</i><sub>0</sub> a square-free number. We prove that any complementing pair in a cyclic group of order <i>p</i><sup><i>a</i></sup><i>n</i><sub>0</sub> is quasi-periodic, with one component decomposable by the the subgroup of order <i>p</i>. The proof is by induction and reduction since the presence of the square-free factor <i>n</i><sub>0</sub> allows us to perform a Tijdeman decomposition. We also give an explicit example to show that <span>\\(\\mathbb{Z}_{72}\\)</span> is the smallest cyclic group that fails to have the strong Tijdeman property. </p></div>","PeriodicalId":50894,"journal":{"name":"Acta Mathematica Hungarica","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Hungarica","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10474-023-01361-3","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
Let pa be a prime power and n0 a square-free number. We prove that any complementing pair in a cyclic group of order pan0 is quasi-periodic, with one component decomposable by the the subgroup of order p. The proof is by induction and reduction since the presence of the square-free factor n0 allows us to perform a Tijdeman decomposition. We also give an explicit example to show that \(\mathbb{Z}_{72}\) is the smallest cyclic group that fails to have the strong Tijdeman property.
期刊介绍:
Acta Mathematica Hungarica is devoted to publishing research articles of top quality in all areas of pure and applied mathematics as well as in theoretical computer science. The journal is published yearly in three volumes (two issues per volume, in total 6 issues) in both print and electronic formats. Acta Mathematica Hungarica (formerly Acta Mathematica Academiae Scientiarum Hungaricae) was founded in 1950 by the Hungarian Academy of Sciences.