{"docId":394,"paperId":394,"url":"https:\/\/dmtcs.episciences.org\/394","doi":"10.46298\/dmtcs.394","journalName":"Discrete Mathematics & Theoretical Computer Science","issn":"","eissn":"1365-8050","volume":[{"vid":82,"name":"Vol. 9 no. 2"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-00966530","repositoryVersion":1,"repositoryLink":"https:\/\/hal.science\/hal-00966530v1","dateSubmitted":"2015-03-26 16:19:24","dateAccepted":"2015-06-09 14:46:31","datePublished":"2007-01-01 08:00:00","titles":{"en":"Addition and multiplication of beta-expansions in generalized Tribonacci base"},"authors":["Ambro\u017e, Petr","Mas\u00e1kov\u00e1, Zuzana","Pelantov\u00e1, Edita"],"abstracts":{"en":"We study properties of \u03b2-numeration systems, where \u03b2 > 1 is the real root of the polynomial x3 - mx2 - x - 1, m \u2208 \u2115, m \u2265 1. We consider arithmetic operations on the set of \u03b2-integers, i.e., on the set of numbers whose greedy expansion in base \u03b2 has no fractional part. We show that the number of fractional digits arising under addition of \u03b2-integers is at most 5 for m \u2265 3 and 6 for m = 2, whereas under multiplication it is at most 6 for all m \u2265 2. We thus generalize the results known for Tribonacci numeration system, i.e., for m = 1. We summarize the combinatorial properties of infinite words naturally defined by \u03b2-integers. We point out the differences between the structure of \u03b2-integers in cases m = 1 and m \u2265 2."},"keywords":["[INFO.INFO-DM] Computer Science [cs]\/Discrete Mathematics [cs.DM]"]}