Abstract
We present a survey of results concerning regular sequences and related objects. Regular sequences were defined in the early 1990s by Allouche and Shallit as a combinatorially, algebraically, and analytically interesting generalization of automatic sequences. In this chapter, after an historical introduction, we follow the development from automatic sequences to regular sequences, and their associated generating functions, to Mahler functions. We then examine size and growth properties of regular sequences. The last half of the chapter focuses on the algebraic, analytic, and Diophantine properties of Mahler functions. In particular, we survey the rational-transcendental dichotomies of Mahler functions, due to Bézivin, and of regular numbers, due to Bell, Bugeaud, and Coons.
The research of M. Coons was supported in part by Australian Research Council grant DE140100223. Lukas Spiegelhofer was supported by the Austrian Science Fund (FWF), projects F5502-N26 and F5505-N26, which are part of the Special Research Program “Quasi Monte Carlo Methods: Theory and Applications”, and also by the ANR–FWF joint project MuDeRa (Multiplicativity, Determinism and Randomness). The authors thank the Erwin Schrödinger Institute for Mathematics and Physics where part of this chapter was written during the workshop on “Normal Numbers: Arithmetic, Computational and Probabilistic Aspects.”
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
Two integers k and l are multiplicatively independent provided \(\log k/\log l\) is irrational.
- 3.
This result is inherent in the work of Cobham. In the 1980s, Loxton and van der Poorten [389] claimed to have proved that an automatic number is either rational or transcendental, but a few unresolvable flaws were found in their argument. This is why their name is associated with the conjecture.
- 4.
We make no comment on the randomness properties of integer sequences, but will be content with their generality as is.
- 5.
Allouche and Shallit gave a more general treatment for sequences taking values in Noetherian rings. In our applications, the most important settings are those of the integers and complex numbers, depending on the type of result presented. For our purposes, for results on sequences and numbers, the integers will be the standard setting, and for results on power series those with complex coefficients will be the most important.
References
Adamczewski, B., Bell, J.P.: A problem around Mahler functions. Ann. Sc. Norm. Super. Pisa. (2013, to appear). ArXiv:1303.2019
Adamczewski, B., Bugeaud, Y.: On the complexity of algebraic numbers. I. Expansions in integer bases. Ann. Math. (2) 165(2), 547–565 (2007)
Adamczewski, B., Faverjon, C.: Méthode de Mahler: relations linéaires, transcendance et application aux nombres automatiques. Proc. Lond. Math. Soc. 115, 55–90 (2017)
Allouche, J.-P., Shallit, J.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, Cambridge (2003)
Allouche, J.-P., Shallit, J.O.: The ring of k-regular sequences. Theor. Comput. Sci. 98, 163–197 (1992)
Becker, P.G.: Effective measures for algebraic independence of the values of Mahler type functions. Acta Arith. 58(3), 239–250 (1991)
Becker, P.G.: k-regular power series and Mahler-type functional equations. J. Number Theory 49(3), 269–286 (1994)
Bell, J.P., Bugeaud, Y., Coons, M.: Diophantine approximation of Mahler numbers. Proc. Lond. Math. Soc. (3) 110(5), 1157–1206 (2015)
Bell, J.P., Coons, M.: Transcendence tests for Mahler functions. Proc. Am. Math. Soc. 145(3), 1061–1070 (2017)
Bell, J.P., Coons, M., Hare, K.G.: The minimal growth of a k-regular sequence. Bull. Aust. Math. Soc. 90(2), 195–203 (2014)
Bell, J.P., Coons, M., Rowland, E.: The rational-transcendental dichotomy of Mahler functions. J. Integer Seq. 16(2), 11 (2013). Article 13.2.10
Bézivin, J.P.: Sur une classe d’équations fonctionnelles non linéaires. Funkcial. Ekvac. 37(2), 263–271 (1994)
Blondel, V.D., Nesterov, Y., Theys, J.: On the accuracy of the ellipsoid norm approximation of the joint spectral radius. Linear Algebra Appl. 394, 91–107 (2005)
Blondel, V.D., Theys, J., Vladimirov, A.A.: An elementary counterexample to the finiteness conjecture. SIAM J. Matrix Anal. Appl. 24(4), 963–970 (electronic) (2003)
Borel, É.: Les probabilités dénombrables et leurs applications arithmétiques. Rendiconti Circ. Mat. Palermo 27, 247–271 (1909)
Bousch, T., Mairesse, J.: Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture. J. Am. Math. Soc. 15(1), 77–111 (electronic) (2002)
Bugeaud, Y.: Expansions of algebraic numbers. In: Four Faces of Number Theory. EMS Series of Lectures in Mathematics, pp. 31–75. European Mathematical Society, Zürich (2015)
Bundschuh, P.: Algebraic independence of infinite products and their derivatives. In: Number Theory and Related Fields. Springer Proceedings in Mathematics and Statistics, vol. 43, pp. 143–156. Springer, New York (2013)
Calkin, N.J., Wilf, H.S.: Binary partitions of integers and Stern-Brocot-like trees. unpublished pp. updated version August 5, 2009, 19 pp. (1998)
Cobham, A.: On the base-dependence of sets of numbers recognizable by finite automata. Math. Syst. Theory 3, 186–192 (1969)
Coons, M.: Regular sequences and the joint spectral radius. Int. J. Found. Comput. Sci. 28(2), 135–140 (2017)
Coons, M.: Zero order estimate for Mahler functions. N. Z. J. Math. 46, 83–88 (2016)
Coons, M., Tyler, J.: The maximal order of Stern’s diatomic sequence. Mosc. J. Comb. Number Theory 4(3), 3–14 (2014)
Daubechies, I., Lagarias, J.C.: Sets of matrices all infinite products of which converge. Linear Algebra Appl. 161, 227–263 (1992)
Dumas, P.: Récurrences mahlériennes, suites automatiques, études asymptotiques. Institut National de Recherche en Informatique et en Automatique (INRIA), Rocquencourt (1993). Thèse, Université de Bordeaux I, Talence, 1993
Dumas, P.: Joint spectral radius, dilation equations, and asymptotic behavior of radix-rational sequences. Linear Algebra Appl. 438(5), 2107–2126 (2013)
Dumas, P.: Asymptotic expansions for linear homogeneous divide-and-conquer recurrences: algebraic and analytic approaches collated. Theor. Comput. Sci. 548, 25–53 (2014)
Eilenberg, S.: Automata, Languages, and Machines, vol. A. Academic Press, New York (1974)
Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)
Hare, K.G., Morris, I.D., Sidorov, N., Theys, J.: An explicit counterexample to the Lagarias-Wang finiteness conjecture. Adv. Math. 226(6), 4667–4701 (2011)
Hartmanis, J., Stearns, R.E.: On the computational complexity of algorithms. Trans. Am. Math. Soc. 117, 285–306 (1965)
Jungers, R.: The Joint Spectral Radius, Theory and Applications. Lecture Notes in Control and Information Sciences, vol. 385. Springer, Berlin (2009)
Jungers, R.M., Blondel, V.D.: On the finiteness property for rational matrices. Linear Algebra Appl. 428(10), 2283–2295 (2008)
Kozyakin, V.S.: A dynamical systems construction of a counterexample to the finiteness conjecture. In: Proceedings of the 44th IEEE Conference on Decision and Control, European Control Conference, pp. 2338–2343 (2005)
Lagarias, J.C., Wang, Y.: The finiteness conjecture for the generalized spectral radius of a set of matrices. Linear Algebra Appl. 214, 17–42 (1995)
Lansing, J.: Distribution of values of the binomial coefficients and the Stern sequence. J. Integer Seq. 16(3), Article 13.3.7, 10 (2013)
Loxton, J.H.: A method of Mahler in transcendence theory and some of its applications. Bull. Aust. Math. Soc. 29(1), 127–136 (1984)
Loxton, J.H., van der Poorten, A.J.: Arithmetic properties of automata: regular sequences. J. Reine Angew. Math. 392, 57–69 (1988)
Mahler, K.: Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen. Math. Ann. 101(1), 342–366 (1929)
Mahler, K.: Uber das Verschwinden von Potenzreihen mehrerer Veränderlichen in speziellen Punktfolgen. Math. Ann. 103(1), 573–587 (1930)
Mahler, K.: Remarks on a paper by W. Schwarz. J. Number Theory 1, 512–521 (1969)
McNaughton, R., Zalcstein, Y.: The Burnside problem for semigroups. J. Algebra 34, 292–299 (1975)
Nesterenko, Y.V.: Estimate of the orders of the zeroes of functions of a certain class, and their application in the theory of transcendental numbers. Izv. Akad. Nauk SSSR Ser. Mat. 41(2), 253–284, 477 (1977)
Nesterenko, Y.V.: Algebraic independence of algebraic powers of algebraic numbers. Mat. Sb. (N.S.) 123(165)(4), 435–459 (1984)
Nikishin, E.M., Sorokin, V.N.: Rational approximations and orthogonality. Translations of Mathematical Monographs, vol. 92. American Mathematical Society, Providence, RI (1991). Translated from the Russian by Ralph P. Boas
Nishioka, K.: Algebraic independence measures of the values of Mahler functions. J. Reine Angew. Math. 420, 203–214 (1991)
Randé, B.: Équations fonctionnelles de Mahler et applications aux suites p-régulières. Institut National de Recherche en Informatique et en Automatique (INRIA), Rocquencourt (1992). Thèse, Université de Bordeaux I, Talence, 1992
Reznick, B.: Some binary partition functions. In: Analytic Number Theory (Allerton Park, IL, 1989). Progress in Mathematics, vol. 85, pp. 451–477. Birkhäuser, Boston, MA (1990)
Rota, G.C., Strang, G.: A note on the joint spectral radius. Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math. 22, 379–381 (1960)
Roth, K.F.: Rational approximations to algebraic numbers. Mathematika 2, 1–20 (1955). corrigendum, 168
Schlickewei, H.P.: The \({\mathfrak p}\)-adic Thue-Siegel-Roth-Schmidt theorem. Arch. Math. (Basel) 29(3), 267–270 (1977)
Töpfer, T.: Zero order estimates for functions satisfying generalized functional equations of Mahler type. Acta Arith. 85(1), 1–12 (1998)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Coons, M., Spiegelhofer, L. (2018). Number Theoretic Aspects of Regular Sequences. In: Berthé, V., Rigo, M. (eds) Sequences, Groups, and Number Theory. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-69152-7_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-69152-7_2
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-69151-0
Online ISBN: 978-3-319-69152-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)