A1 Journal article – refereed

Cellular automata and powers of p/q

List of Authors: Jarkko Kari, Johan Kopra

Publisher: EDP Sciences

Publication year: 2017

Journal: RAIRO: Informatique Théorique et Applications / RAIRO: Theoretical Informatics and Applications

Journal name in source: RAIRO - Theoretical Informatics and Applications

Volume number: 51

Issue number: 4

Number of pages: 14

ISSN: 0988-3754

DOI: http://dx.doi.org/10.1051/ita/2017014


We consider one-dimensional cellular automata Fp,q which multiply numbers by pq in base pq for relatively prime integers p and q. By studying the structure of traces with respect to Fp,q we show that for p ≥ 2q – 1 (and then as a simple corollary for p > q > 1) there are arbitrarily small finite unions of intervals which contain the fractional parts of the sequence ξ(pq)n, (n = 0, 1, 2, …) for some ξ > 0. To the other direction, by studying the measure theoretical properties of Fp,q, we show that for p > q > 1 there are finite unions of intervals approximating the unit interval arbitrarily well which don’t contain the fractional parts of the whole sequence ξ(pq)n for any ξ > 0.

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Last updated on 2021-24-06 at 08:54