On the rational approximation to the Thue–Morse–Mahler numbers
Annales de l'Institut Fourier, Volume 61 (2011) no. 5, p. 2065-2076

Let (t k ) k0 be the Thue–Morse sequence on {0,1} defined by t 0 =0, t 2k =t k and t 2k+1 =1-t k for k0. Let b2 be an integer. We establish that the irrationality exponent of the Thue–Morse–Mahler number k0 t k b -k is equal to 2.

Soit (t k ) k0 la suite de Thue–Morse définie sur {0,1} par t 0 =0, t 2k =t k et t 2k+1 =1-t k pour k0. Soit b2 un entier rationnel. Nous démontrons que l’exposant d’irrationalité du nombre de Thue–Morse–Mahler k0 t k b -k est égal à 2.

DOI : https://doi.org/10.5802/aif.2666
Classification:  11J04,  11J82
Keywords: Irrationality measure, Thue–Morse sequence, Padé approximant
@article{AIF_2011__61_5_2065_0,
     author = {Bugeaud, Yann},
     title = {On the rational approximation to the Thue--Morse--Mahler numbers},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {5},
     year = {2011},
     pages = {2065-2076},
     doi = {10.5802/aif.2666},
     mrnumber = {2961848},
     zbl = {1271.11074},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_5_2065_0}
}
Bugeaud, Yann. On the rational approximation to the Thue–Morse–Mahler numbers. Annales de l'Institut Fourier, Volume 61 (2011) no. 5, pp. 2065-2076. doi : 10.5802/aif.2666. http://www.numdam.org/item/AIF_2011__61_5_2065_0/

[1] Adamczewski, Boris On the expansion of some exponential periods in an integer base, Math. Ann., Tome 346 (2010) no. 1, pp. 107-116 | Article | MR 2558889 | Zbl pre05661736

[2] Adamczewski, Boris; Cassaigne, Julien Diophantine properties of real numbers generated by finite automata, Compos. Math., Tome 142 (2006) no. 6, pp. 1351-1372 | Article | MR 2278750 | Zbl 1134.11011

[3] Adamczewski, Boris; Rivoal, Tanguy Irrationality measures for some automatic real numbers, Math. Proc. Cambridge Philos. Soc., Tome 147 (2009) no. 3, pp. 659-678 | Article | MR 2557148 | Zbl 1205.11080

[4] Adams, William W.; Davison, J. L. A remarkable class of continued fractions, Proc. Amer. Math. Soc., Tome 65 (1977) no. 2, pp. 194-198 | Article | MR 441879 | Zbl 0366.10027

[5] Allouche, J.-P.; Peyrière, J.; Wen, Z.-X.; Wen, Z.-Y. Hankel determinants of the Thue-Morse sequence, Ann. Inst. Fourier (Grenoble), Tome 48 (1998) no. 1, pp. 1-27 | Article | Numdam | MR 1614914 | Zbl 0974.11010

[6] Allouche, J.-P.; Shallit, J. O. Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press (2003) | MR 1997038 | Zbl 1086.11015

[7] Amou, Masaaki Approximation to certain transcendental decimal fractions by algebraic numbers, J. Number Theory, Tome 37 (1991) no. 2, pp. 231-241 | Article | MR 1092608 | Zbl 0718.11030

[8] Baker, George A. Jr.; Graves-Morris, Peter Padé approximants, Cambridge University Press, Cambridge, Encyclopedia of Mathematics and its Applications, Tome 59 (1996) | Article | MR 1383091 | Zbl 0923.41001

[9] Berthé, Valérie; Holton, Charles; Zamboni, Luca Q. Initial powers of Sturmian sequences, Acta Arith., Tome 122 (2006) no. 4, pp. 315-347 | Article | MR 2234421 | Zbl 1117.37005

[10] Brezinski, Claude Padé-type approximation and general orthogonal polynomials, Birkhäuser Verlag, Basel, International Series of Numerical Mathematics, Tome 50 (1980) | MR 561106 | Zbl 0418.41012

[11] Bugeaud, Yann Diophantine approximation and Cantor sets, Math. Ann., Tome 341 (2008) no. 3, pp. 677-684 | Article | MR 2399165 | Zbl 1163.11056

[12] Bugeaud, Yann; Krieger, D.; Shallit, J. Morphic and Automatic Words: Maximal Blocks and Diophantine Approximation (Preprint) | Zbl pre05908266

[13] Bundschuh, Peter Über eine Klasse reeller transzendenter Zahlen mit explizit angebbarer g-adischer und Kettenbruch-Entwicklung, J. Reine Angew. Math., Tome 318 (1980), pp. 110-119 | Article | MR 579386 | Zbl 0425.10038

[14] Dekking, Michel Transcendance du nombre de Thue-Morse, C. R. Acad. Sci. Paris Sér. A-B, Tome 285 (1977) no. 4, p. A157-A160 | MR 457363 | Zbl 0362.10028

[15] Levesley, Jason; Salp, Cem; Velani, Sanju L. On a problem of K. Mahler: Diophantine approximation and Cantor sets, Math. Ann., Tome 338 (2007) no. 1, pp. 97-118 | Article | MR 2295506 | Zbl 1115.11040

[16] Mahler, Kurt Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Math. Ann., Tome 101 (1929) no. 1, pp. 342-366 | Article | MR 1512537

[17] Shallit, Jeffrey Simple continued fractions for some irrational numbers, J. Number Theory, Tome 11 (1979) no. 2, pp. 209-217 | Article | MR 535392 | Zbl 0404.10003

[18] Sloane, N. J. A. The on-line encyclopedia of integer sequences, Notices Amer. Math. Soc., Tome 50 (2003) no. 8, pp. 912-915 | MR 1992789 | Zbl 1044.11108