{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:45Z","timestamp":1753893825565,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We prove a result concerning the possible orders of a basis for the cyclic group ${\\Bbb Z}_n$, namely: For each $k \\in {\\Bbb N}$ there exists a constant $c_k &gt; 0$ such that, for all $n \\in {\\Bbb N}$, if $A \\subseteq {\\Bbb Z}_n$ is a basis of order greater than $n\/k$, then the order of $A$ is within $c_k$ of $n\/l$ for some integer $l \\in [1,k]$. The proof makes use of various results in additive number theory concerning the growth of sumsets.  Additionally, exact results are summarized for the possible basis orders greater than $n\/4$ and less than $\\sqrt{n}$.  An equivalent problem in graph theory is discussed, with applications.<\/jats:p>","DOI":"10.37236\/351","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:13:03Z","timestamp":1578715983000},"source":"Crossref","is-referenced-by-count":0,"title":["On the Possible Orders of a Basis for a Finite Cyclic Group"],"prefix":"10.37236","volume":"17","author":[{"given":"Peter","family":"Dukes","sequence":"first","affiliation":[]},{"given":"Peter","family":"Hegarty","sequence":"additional","affiliation":[]},{"given":"Sarada","family":"Herke","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2010,5,25]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r79\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v17i1r79\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T23:51:21Z","timestamp":1579305081000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v17i1r79"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,5,25]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2010,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/351","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2010,5,25]]},"article-number":"R79"}}