×

On the expansion of some exponential periods in an integer base. (English) Zbl 1247.11095

The author derives a lower bound for the subword complexity of the base-\(b\) expansion (\(b\geq 2\)) of all real numbers whose irrationality exponent is equal to 2. This provides a generalization of a theorem due to S. Ferenczi and C. Mauduit [J. Number Theory 67, 146–161 (1997; Zbl 0895.11029)]. As a consequence, the author obtains the first lower bound for the subword complexity of the number \(e\) and of some other transcendental exponential periods, including \[ \begin{aligned} &e^a \quad (a\in\mathbb{Q}^\times),\\ &\tan(1/a), \;\sqrt{a}\tan(1/\sqrt{a}),\;(1/\sqrt{a})\tan(1/\sqrt{a})\quad (a\in\mathbb{N}^\times),\\ &\tanh (2/a)\quad (a\in\mathbb{N}^\times),\\ &\sqrt{v}/\sqrt{u})\tanh (1/\sqrt{uv})\quad (u,v\in\mathbb{N}, uv\neq 0),\\ &\frac{J_{(p/q)+1}(2/q)}{J_{p/q}(2/q)}\text{ and }\frac{I_{(p/q)+1}(2/q)}{I_{p/q}(2/q)}\quad (p/q\in\mathbb{Q}), \end{aligned} \] where \(J_\lambda(z)\) and \(I_\lambda(z)\) denote the Bessel function of the first kind and the modified Bessel function of the first kind, respectively (see for instance [T. Komatsu, Czech. Math. J. 57, 919–932 (2007; Zbl 1163.11009)]).

MSC:

11J82 Measures of irrationality and of transcendence
11A63 Radix representation; digital problems
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adamczewski B., Bugeaud Y.: On the complexity of algebraic numbers I. Expansions in integer bases. Ann. Math. 165, 547–565 (2007) · Zbl 1195.11094
[2] Adamczewski B., Bugeaud Y.: Dynamics for {\(\beta\)}-shifts and Diophantine approximation. Ergod. Th. Dyn. Sys. 27, 1695–1710 (2007) · Zbl 1140.11035
[3] Adamczewski, B., Bugeaud, Y.: Nombres réels de complexité sous-linéaire : mesures d’irrationalité et de transcendance, Preprint available at http://math.univ-lyon1.fr/\(\sim\)adamczew/ClasseCL.pdf
[4] Adamczewski B., Bugeaud Y., Luca F.: Sur la complexité des nombres algébriques. C. R. Acad. Sci. Paris 339, 11–14 (2004) · Zbl 1119.11019
[5] Adamczewski B., Cassaigne J.: Diophantine properties of real numbers generated by finite automata. Compos. Math. 142, 1351–1372 (2006) · Zbl 1134.11011
[6] Allouche J.-P.: Nouveaux résultats de transcendance de réels à développements non aléatoire. Gaz. Math. 84, 19–34 (2000)
[7] Allouche J.-P., Shallit J.O.: Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, London (2003) · Zbl 1086.11015
[8] Baker A.: Approximations to the logarithms of certain rational numbers. Acta Arith. 10, 315–323 (1964) · Zbl 0201.37603
[9] Berthé V., Holton C., Zamboni L.Q.: Initial powers of Sturmian words. Acta Arith. 122, 315–347 (2006) · Zbl 1117.37005
[10] Borel É.: Les probabilités dénombrables et leurs applications arithmétiques. Rend. Circ. Mat. Palermo 27, 247–271 (1909) · JFM 40.0283.01
[11] Bundschuh P.: Irrationalitätsmaße für e a , a 0 rational oder Liouville-Zahl. Math. Ann. 129, 229–242 (1971) · Zbl 0209.34702
[12] Cassaigne, J.: Sequences with grouped factors. In: DLT’97, Developments in Language Theory III, Thessaloniki, Aristotle University of Thessaloniki, pp. 211–222 (1998)
[13] Ferenczi S., Mauduit Ch.: Transcendence of numbers with a low complexity expansion. J. Number Theory 67, 146–161 (1997) · Zbl 0895.11029
[14] Gheorghiciuc I.: The subword complexity of a class of infinite binary words. Adv. Appl. Math. 39, 237–259 (2007) · Zbl 1117.68059
[15] Khintchine A.Ya.: Continued Fractions. University of Chicago Press, Chicago (1964) · Zbl 0117.28601
[16] Komatsu T.: Hurwitz continued fractions with confluent hypergeometric functions. Czechoslovak Math. J. 57, 919–932 (2007) · Zbl 1163.11009
[17] Kontsevich, M., Zagier, D.: Periods. In: Mathematics unlimited–2001 and beyond, pp. 771–808. Springer, Heidelberg (2001) · Zbl 1039.11002
[18] Morse M., Hedlund G.A.: Symbolic dynamics. Am. J. Math. 60, 815–866 (1938) · JFM 64.0798.04
[19] Ridout D.: Rational approximations to algebraic numbers. Mathematika 4, 125–131 (1957) · Zbl 0079.27401
[20] Roth K.F.: Rational approximations to algebraic numbers. Mathematika 2, 1–20 (1955) [corrigendum, 169] · Zbl 0064.28501
[21] Shallit J.: Simple continued fractions for some irrational numbers. J. Number Theory 11, 209–217 (1979) · Zbl 0404.10003
[22] Tasoev, B.G.: Rational approximations to certain numbers. Mat. Zametki 67 931–937 (2000) [translation in Math. Notes 67, 786–791 (2000)] · Zbl 0989.11033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.