×

On the transcendence of real numbers with a regular expansion. (English) Zbl 1052.11052

This paper is concerned with the long-standing problem: “How regular or random is the \(b\)-ary expansion of an algebraic irrational number?” where \(b\) is any integer greater than one. It is well known that the \(b\)-ary expansion of a rational number is ultimately periodic. It has been conjectured that the \(b\)-ary expansion of an irrational number is totally random, and several results show that if such expansion is “too regular”, then the number is transcendental. Transcendence results have been obtained from a recent purely combinatorial condition formulated by Ferenczi and Mauduit. In particular, this condition has been used to obtain the transcendence of real numbers whose \(b\)-ary expansion is a Sturmian sequence.
Here, the authors generalize this result by using the Ferenczi-Mauduit condition to prove the transcendence of real numbers whose \(b\)-ary expansion is the coding of an irrational rotation on the circle with respect to a partition in two intervals or arises from a non-periodic three-interval exchange transformation.

MSC:

11J81 Transcendence (general theory)
11B83 Special sequences and polynomials
68R15 Combinatorics on words
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Adamczewski, B., Codages de rotations et phénomènes d’autosimilarité, J. théor. nombres Bordeaux, 14, 351-386, (2002) · Zbl 1113.37003
[2] Adams, W.W.; Davison, J.L., A remarkable class of continued fractions, Proc. amer. math. soc., 65, 2, 194-198, (1977) · Zbl 0366.10027
[3] Allouche, J.-P., Nouveaux résultats de transcendance de réels à développement non aléatoire, Gaz. math., 84, 19-34, (2000) · Zbl 1388.11046
[4] Allouche, J.-P.; Zamboni, L.Q., Algebraic irrational binary numbers cannot be fixed points of non-trivial constant length or primitive morphisms, J. number theory, 69, 119-124, (1998) · Zbl 0918.11016
[5] Böhmer, P.E., Über die transzendenz gewisser dyadischer brüche, Math. ann., 96, 367-377, (1926), (Erratum 96 (1926) 735) · JFM 52.0188.02
[6] Borel, É., Sur LES chiffres décimaux de \(2\) et divers problèmes de probabilités en chaı̂ne, C. R. acad. sci. Paris, 230, 591-593, (1950) · Zbl 0035.08302
[7] J. Cassaigne, Sequences with grouped factors, Developments in Language Theory III, Publications of the Aristotle University of Thessaloniki, 1998, pp. 211-222.
[8] Danilov, L.V., Certain classes of transcendental numbers, Mat. zametki, 12, 149-154, (1972)
[9] Davison, J.L., A series and its associated continued fraction, Proc. amer. math. soc., 63, 1, 29-32, (1977) · Zbl 0326.10030
[10] Dekking, M., Transcendance du nombre de thue-Morse, C. R. acad. sci. Paris Sér. A-B, 285, 4, A157-A160, (1977)
[11] Ferenczi, S.; Mauduit, C., Transcendence of numbers with a low complexity expansion, J. number theory, 67, 146-161, (1997) · Zbl 0895.11029
[12] Keane, M., Interval exchange transformations, Math. Z., 141, 25-31, (1975) · Zbl 0278.28010
[13] Loxton, J.H.; van der Poorten, A.J., Arithmetic properties of automataregular sequences, J. reine angew. math., 392, 57-69, (1988) · Zbl 0656.10033
[14] Mahler, K., Arithmetische eigenschaften der Lösungen einer klasse von funktionalgleichungen, Math. ann., 101, 342-366, (1929), (Corrigendum 103 (1930) 532) · JFM 55.0115.01
[15] Mahler, K., Arithmetische eigenschaften einer klasse von dezimalbrüchen, Proc. konin. nederl. akad. wetensch. ser. A, 40, 421-428, (1937) · JFM 63.0156.01
[16] K. Mahler, Lectures on Diophantine approximations, part I: g-adic numbers and Roth’s theorem, University of Notre Dame, 1961. · Zbl 0158.29903
[17] Ridout, D., Rational approximations to algebraic numbers, Mathematika, 4, 125-131, (1957) · Zbl 0079.27401
[18] Risley, R.N.; Zamboni, L.Q., A generalization of Sturmian sequencescombinatorial structure and transcendence, Acta arith., 95, 167-184, (2000) · Zbl 0953.11007
[19] Rote, G., Sequences with subword complexity 2n, J. number theory, 46, 196-213, (1994) · Zbl 0804.11023
[20] Roth, K.F., Rational approximations to algebraic numbers, Mathematika, 2, 1-20, (1955), (Corrigendum 2 (1955) 168) · Zbl 0064.28501
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.