no. 4
Cellular automata and powers of p∕q
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Special issue dedicated to the 16th "Journées Montoises d’Informatique Théorique", Tome 51 (2017) no. 4, pp. 191-204

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

DOI : 10.1051/ita/2017014
Classification : 11J71, 37A25, 68Q80
Keywords: Distribution modulo 1, Z-numbers, cellular automata, ergodicity, strongly mixing

Kari, Jarkko 1 ; Kopra, Johan 1

1 
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Kari, Jarkko; Kopra, Johan. Cellular automata and powers of p∕q. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Special issue dedicated to the 16th "Journées Montoises d’Informatique Théorique", Tome 51 (2017) no. 4, pp. 191-204. doi: 10.1051/ita/2017014

[1] S. Akiyama, Mahler’s Z-number and 3/2 number system. Unif. Distrib. Theory 3 (2008) 91–99. | MR | Zbl

[2] S. Akiyama, Ch. Frougny and J. Sakarovitch, Powers of rationals modulo 1 and rational base number systems. Isr. J. Math. 168 (2008) 53–91. | MR | Zbl | DOI

[3] A. Dubickas, On the powers of 3/2 and other rational numbers. Math. Nachr. 281 (2008) 951–958. | MR | Zbl | DOI

[4] L. Flatto, J.C. Lagarias and A.D. Pollington, On the range of fractional parts {ξ(p∕q)n }. Acta Arith. 70 (1995) 125–147. | MR | Zbl | DOI

[5] G. Hedlund, Endomorphisms and automorphisms of shift dynamical systems. Math. Syst. Theory 3 (1969) 320–375. | MR | Zbl | DOI

[6] J. Kari, Cellular automata, the Collatz conjecture and powers of 3/2, in Developments in Language Theory. Vol. 7410 of Lecture Notes in Computer Science (2012) 40–49. | MR | Zbl | DOI

[7] J. Kari, Universal pattern generation by cellular automata. Theor. Comput. Sci. 429 (2012) 180–184. | MR | Zbl | DOI

[8] K. Mahler, An unsolved problem on the powers of 3/2. J. Aust. Math. Soc. 8 (1968) 313–321. | MR | Zbl | DOI

[9] W. Rudin, Real and Complex Analysis. McGraw-Hill Book Company, New York (1966). | MR | Zbl

[10] P. Walters, An Introduction to Ergodic Theory. Springer-Verlag, New York (1982). | MR | DOI

[11] H. Weyl, Über die Gleichverteilung von Zahlen mod. Eins. Math. Ann. 77 (1916) 313–352. | MR | JFM | DOI

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