{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:55Z","timestamp":1753893835547,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The Thue-Morse set $\\mathcal{T}$ is the set of those non-negative integers whose binary expansions have an even number of $1$'s. The name of this set comes from the fact that its characteristic sequence is given by the famous Thue-Morse word\r\n$${\\tt 0110100110010110\\cdots},$$\r\nwhich is the fixed point starting with ${\\tt 0}$ of the word morphism ${\\tt 0\\mapsto 01}$, ${\\tt 1\\mapsto 10}$. The numbers in $\\mathcal{T}$ are commonly called the evil numbers. We obtain an exact formula for the state complexity of the set $m\\mathcal{T}+r$ (i.e. the number of states of its minimal automaton) with respect to any base $b$ which is a power of $2$. Our proof is constructive and we are able to explicitly provide the minimal automaton of the language of all $2^p$-expansions of the set of integers $m\\mathcal{T}+r$ for any positive integers $p$ and $m$ and any remainder $r\\in\\{0,\\ldots,m{-}1\\}$. The proposed method is general for any $b$-recognizable set of integers.<\/jats:p>","DOI":"10.37236\/9068","type":"journal-article","created":{"date-parts":[[2021,7,2]],"date-time":"2021-07-02T04:41:53Z","timestamp":1625200913000},"source":"Crossref","is-referenced-by-count":0,"title":["Minimal Automaton for Multiplying and Translating the Thue-Morse Set"],"prefix":"10.37236","volume":"28","author":[{"given":"\u00c9milie","family":"Charlier","sequence":"first","affiliation":[]},{"given":"C\u00e9lia","family":"Cisternino","sequence":"additional","affiliation":[]},{"given":"Adeline","family":"Massuir","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2021,7,2]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i3p12\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i3p12\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,2]],"date-time":"2021-07-02T04:41:54Z","timestamp":1625200914000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v28i3p12"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,2]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,7,1]]}},"URL":"https:\/\/doi.org\/10.37236\/9068","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2021,7,2]]},"article-number":"P3.12"}}