{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,15]],"date-time":"2026-10-15T13:24:53Z","timestamp":1792070693383,"version":"4.4.0"},"reference-count":11,"publisher":"EDP Sciences","issue":"4","license":[{"start":{"date-parts":[[2018,1,31]],"date-time":"2018-01-31T00:00:00Z","timestamp":1517356800000},"content-version":"vor","delay-in-days":122,"URL":"https:\/\/www.edpsciences.org\/en\/authors\/copyright-and-licensing"}],"funder":[{"DOI":"10.13039\/501100002341","name":"Academy of Finland","doi-asserted-by":"publisher","award":["296018"],"award-info":[{"award-number":["296018"]}],"id":[{"id":"10.13039\/501100002341","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Theor. Inf. Appl."],"accepted":{"date-parts":[[2017,12,1]]},"published-print":{"date-parts":[[2017,10]]},"abstract":"We consider one-dimensional cellular automata F<\/jats:italic>p<\/jats:italic>,q<\/jats:italic><\/jats:sub> which multiply numbers by p<\/jats:italic>\u2215q<\/jats:italic> in base pq<\/jats:italic> for relatively prime integers p<\/jats:italic> and q<\/jats:italic>. By studying the structure of traces with respect to F<\/jats:italic>p<\/jats:italic>,q<\/jats:italic><\/jats:sub> we show that for p<\/jats:italic> \u2265 2q<\/jats:italic>\u00a0\u2013 1 (and then as a simple corollary for p<\/jats:italic> > q<\/jats:italic> > 1) there are arbitrarily small finite unions of intervals which contain the fractional parts of the sequence \u03be<\/jats:italic>(p<\/jats:italic>\u2215q<\/jats:italic>)n<\/jats:italic><\/jats:sup>, (n<\/jats:italic> = 0, 1, 2, \u2026) for some \u03be<\/jats:italic> > 0. To the other direction, by studying the measure theoretical properties of F<\/jats:italic>p<\/jats:italic>,q<\/jats:italic><\/jats:sub>, we show that for p<\/jats:italic> > q<\/jats:italic> > 1 there are finite unions of intervals approximating the unit interval arbitrarily well which don\u2019t contain the fractional parts of the whole sequence \u03be<\/jats:italic>(p<\/jats:italic>\u2215q<\/jats:italic>)n<\/jats:italic><\/jats:sup> for any \u03be<\/jats:italic> > 0.<\/jats:p>","DOI":"10.1051\/ita\/2017014","type":"journal-article","created":{"date-parts":[[2018,1,31]],"date-time":"2018-01-31T03:50:38Z","timestamp":1517370638000},"page":"191-204","source":"Crossref","is-referenced-by-count":6,"title":["Cellular automata and powers of p<\/i>\u2215q<\/i>"],"prefix":"10.1051","volume":"51","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0670-6138","authenticated-orcid":false,"given":"Jarkko","family":"Kari","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Johan","family":"Kopra","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"250","published-online":{"date-parts":[[2018,1,31]]},"reference":[{"key":"R1","first-page":"91","volume":"3","author":"Akiyama","year":"2008","journal-title":"Unif. Distrib. Theory"},{"key":"R2","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s11856-008-1056-4","volume":"168","author":"Akiyama","year":"2008","journal-title":"Isr. J. Math."},{"key":"R3","doi-asserted-by":"crossref","first-page":"951","DOI":"10.1002\/mana.200510651","volume":"281","author":"Dubickas","year":"2008","journal-title":"Math. Nachr."},{"key":"R4","doi-asserted-by":"crossref","first-page":"125","DOI":"10.4064\/aa-70-2-125-147","volume":"70","author":"Flatto","year":"1995","journal-title":"Acta Arith."},{"key":"R5","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1007\/BF01691062","volume":"3","author":"Hedlund","year":"1969","journal-title":"Math. Syst. Theory"},{"key":"R6","doi-asserted-by":"crossref","unstructured":"Kari J., Cellular automata, the Collatz conjecture and powers of 3\/2, in Developments in Language Theory. Vol. 7410 of \nLecture Notes in Computer Science \n(2012) 40\u201349.","DOI":"10.1007\/978-3-642-31653-1_5"},{"key":"R7","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/j.tcs.2011.12.037","volume":"429","author":"Kari","year":"2012","journal-title":"Theor. Comput. Sci."},{"key":"R8","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1017\/S1446788700005371","volume":"8","author":"Mahler","year":"1968","journal-title":"J. Aust. Math. Soc."},{"key":"R9","unstructured":"Rudin W., \nReal and Complex Analysis. \nMcGraw-Hill Book Company, \nNew York \n(1966)."},{"key":"R10","doi-asserted-by":"crossref","unstructured":"Walters P., \nAn Introduction to Ergodic Theory. \nSpringer-Verlag, \nNew York \n(1982).","DOI":"10.1007\/978-1-4612-5775-2"},{"key":"R11","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/BF01475864","volume":"77","author":"Weyl","year":"1916","journal-title":"Math. Ann."}],"container-title":["RAIRO - Theoretical Informatics and Applications"],"original-title":[],"link":[{"URL":"https:\/\/www.rairo-ita.org\/10.1051\/ita\/2017014\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,3]],"date-time":"2021-09-03T07:54:12Z","timestamp":1630655652000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.rairo-ita.org\/10.1051\/ita\/2017014"}},"subtitle":[],"editor":[{"given":"J.","family":"Leroy","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"editor"}]},{"given":"M.","family":"Rigo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"editor"}]},{"given":"E.","family":"Charlier","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"editor"}]}],"short-title":[],"issued":{"date-parts":[[2017,10]]},"references-count":11,"journal-issue":{"issue":"4"},"alternative-id":["ita170090"],"URL":"https:\/\/doi.org\/10.1051\/ita\/2017014","relation":{},"ISSN":["0988-3754","1290-385X"],"issn-type":[{"value":"0988-3754","type":"print"},{"value":"1290-385X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,10]]}}}