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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

  • JASON P. BELL (a1), MICHAEL COONS (a2) and KEVIN G. HARE (a3)

Abstract

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.

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      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
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References

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[1]Allouche, J.-P. and Shallit, J., ‘The ring of k-regular sequences’, Theoret. Comput. Sci. 98 (1992), 163197.
[2]Allouche, J.-P. and Shallit, J., Automatic Sequences (Cambridge University Press, Cambridge, 2003).
[3]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.
[4]Cobham, A., ‘On the base-dependence of sets of numbers recognizable by finite automata’, Math. Syst. Theor. 3 (1969), 186192.
[5]Cobham, A., ‘Uniform tag sequences’, Math. Syst. Theor. 6 (1972), 164192.
[6]Erdős, P., ‘Some unsolved problems’, Michigan Math. J. 4 (1957), 291300.
[7]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.
[8]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.
[9]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).
[10]Gowers, T., The Erdős discrepancy problem: focusing on multiplicative functions, http://gowers.wordpress.com/2010/01/30/edp4-focusing-on-multiplicative-functions.
[11]McNaughton, R. and Zalcstein, Y., ‘The Burnside problem for semigroups’, J. Algebra 34 (1975), 292299.
[12]Schlage-Puchta, J.-C., ‘Completely multiplicative automatic functions’, Integers 11 (2011), A31, 8 pages.
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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

  • JASON P. BELL (a1), MICHAEL COONS (a2) and KEVIN G. HARE (a3)

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