{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:26Z","timestamp":1753893866679,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A word is called $\\beta$-free if it has no factors of exponent greater than or equal to $\\beta$. The repetition threshold $\\mbox{RT}(k)$ is the infimum of the set of all $\\beta$ such that there are arbitrarily long $k$-ary $\\beta$-free words (or equivalently, there are $k$-ary $\\beta$-free words of every sufficiently large length, or even every length). These three equivalent definitions of the repetition threshold give rise to three natural definitions of a repetition threshold for circular words. The infimum of the set of all $\\beta$ such that\r\n\r\nthere are arbitrarily long $k$-ary $\\beta$-free circular words is called the weak circular repetition threshold, denoted $\\mbox{CRT}_\\mbox{W}(k)$;\r\nthere are $k$-ary\u00a0$\\beta$-free circular words of every sufficiently large length is called the intermediate circular repetition threshold, denoted $\\mbox{CRT}_\\mbox{I}(k)$;\r\nthere are $k$-ary\u00a0$\\beta$-free circular words of every length is called the strong circular repetition threshold, denoted $\\mbox{CRT}_\\mbox{S}(k)$.\r\n\r\nWe prove that $\\mbox{CRT}_\\mbox{S}(4)=\\tfrac{3}{2}$ and $\\mbox{CRT}_\\mbox{S}(5)=\\tfrac{4}{3}$, confirming a conjecture of Gorbunova and providing the last unknown values of the strong circular repetition threshold. We also prove that $\\mbox{CRT}_\\mbox{S}(3)=\\mbox{CRT}_\\mbox{W}(3)=\\mbox{RT}(3)=\\tfrac{7}{4}$.\r\n\u00a0<\/jats:p>","DOI":"10.37236\/7985","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T02:13:46Z","timestamp":1578622426000},"source":"Crossref","is-referenced-by-count":2,"title":["Circular Repetition Thresholds on Some Small Alphabets: Last Cases of Gorbunova's Conjecture"],"prefix":"10.37236","volume":"26","author":[{"given":"James D.","family":"Currie","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4295-0632","authenticated-orcid":false,"given":"Lucas","family":"Mol","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Narad","family":"Rampersad","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2019,5,31]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i2p31\/7845","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i2p31\/7845","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:12:28Z","timestamp":1579216348000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i2p31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,31]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,4,5]]}},"URL":"https:\/\/doi.org\/10.37236\/7985","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2019,5,31]]},"article-number":"P2.31"}}