{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:23:02Z","timestamp":1758824582304,"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>Recently, Fici, Restivo, Silva, and Zamboni defined a $k$-anti-power to be a word of the form $w_1w_2\\cdots w_k$, where $w_1,w_2,\\ldots,w_k$ are distinct words of the same length. They defined $AP(x,k)$ to be the set of all positive integers $m$ such that the prefix of length $km$ of the word $x$ is a $k$-anti-power. Let ${\\bf t}$ denote the Thue-Morse word, and let $\\mathcal F(k)=AP({\\bf t},k)\\cap(2\\mathbb Z^+-1)$. For $k\\geq 3$, $\\gamma(k)=\\min(\\mathcal F(k))$ and $\\Gamma(k)=\\max((2\\mathbb Z^+-1)\\setminus\\mathcal F(k))$ are well-defined odd positive integers. Fici et al. speculated that $\\gamma(k)$ grows linearly in $k$. We prove that this is indeed the case by showing that $1\/2\\leq\\displaystyle{\\liminf_{k\\to\\infty}}(\\gamma(k)\/k)\\leq 9\/10$ and $1\\leq\\displaystyle{\\limsup_{k\\to\\infty}}(\\gamma(k)\/k)\\leq 3\/2$. In addition, we prove that $\\displaystyle{\\liminf_{k\\to\\infty}}(\\Gamma(k)\/k)=3\/2$ and $\\displaystyle{\\limsup_{k\\to\\infty}}(\\Gamma(k)\/k)=3$. <\/jats:p>","DOI":"10.37236\/6321","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T19:17:45Z","timestamp":1578683865000},"source":"Crossref","is-referenced-by-count":5,"title":["Anti-Power Prefixes of the Thue-Morse Word"],"prefix":"10.37236","volume":"24","author":[{"given":"Colin","family":"Defant","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,2,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p32\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i1p32\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:05:03Z","timestamp":1579237503000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i1p32"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,2,17]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2017,1,20]]}},"URL":"https:\/\/doi.org\/10.37236\/6321","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,2,17]]},"article-number":"P1.32"}}