Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T19:29:17.287Z Has data issue: false hasContentIssue false

THE MINIMAL GROWTH OF A $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}k$-REGULAR SEQUENCE

Published online by Cambridge University Press:  22 May 2014

JASON P. BELL
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Canada email [email protected]
MICHAEL COONS*
Affiliation:
School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, Australia email [email protected]
KEVIN G. HARE
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Canada email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We determine a lower gap property for the growth of an unbounded $\mathbb{Z}$-valued $k$-regular sequence. In particular, if $f:\mathbb{N}\to \mathbb{Z}$ is an unbounded $k$-regular sequence, we show that there is a constant $c>0$ such that $|f(n)|>c\log n$ infinitely often. We end our paper by answering a question of Borwein, Choi and Coons on the sums of completely multiplicative automatic functions.

Type
Research Article
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Allouche, J.-P. and Shallit, J., ‘The ring of k-regular sequences’, Theoret. Comput. Sci. 98 (1992), 163197.CrossRefGoogle Scholar
Allouche, J.-P. and Shallit, J., Automatic Sequences (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
Borwein, P., Choi, S. K. K. and Coons, M., ‘Completely multiplicative functions taking values in {−1, 1}’, Trans. Amer. Math. Soc. 362(12) (2010), 62796291.Google Scholar
Cobham, A., ‘On the base-dependence of sets of numbers recognizable by finite automata’, Math. Syst. Theor. 3 (1969), 186192.CrossRefGoogle Scholar
Cobham, A., ‘Uniform tag sequences’, Math. Syst. Theor. 6 (1972), 164192.Google Scholar
Erdős, P., ‘Some unsolved problems’, Michigan Math. J. 4 (1957), 291300.Google Scholar
Erdős, P., ‘On some of my problems in number theory I would most like to see solved’, in: Number Theory (Ootacamund, 1984), Lecture Notes in Mathematics, 1122 (Springer, Berlin, 1985), 74–84.Google Scholar
Erdős, P., ‘Some applications of probability methods to number theory’, in: Mathematical Statistics and Applications, Vol. B (Bad Tatzmannsdorf, 1983) (Reidel, Dordrecht, 1985), 1–18.Google Scholar
Erdős, P. and Graham, R. L., Old and New Problems and Results in Combinatorial Number Theory, Monographies de L’Enseignement Mathématique [Monographs of L’Enseignement Mathématique] 28 (Université de Genève L’Enseignement Mathématique, Geneva, 1980).Google Scholar
Gowers, T., The Erdős discrepancy problem: focusing on multiplicative functions, http://gowers.wordpress.com/2010/01/30/edp4-focusing-on-multiplicative-functions.Google Scholar
McNaughton, R. and Zalcstein, Y., ‘The Burnside problem for semigroups’, J. Algebra 34 (1975), 292299.Google Scholar
Schlage-Puchta, J.-C., ‘Completely multiplicative automatic functions’, Integers 11 (2011), A31, 8 pages.CrossRefGoogle Scholar