Abstract
Let \((t(m))_{m \ge0}\) be Thue-Morse sequence and \(b>2\) be an integer. In this paper, we prove that the real numbers \(1\), \(\sum_{m=0}^\infty {\frac{t(m^2)}{{b}^{m+1}}}\) and \(\sum_{m=0}^\infty {\frac{t(m^3)}{{b}^{m+1}}}\) are linearly independent over \(\mathbb{Q}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Adamczewski, J. Bell and D. Smertnig, A height gap theorem for coefficients of Mahler functions, J. Eur. Math. Soc, to appear.
B. Adamczewski and Y. Bugeaud, On the complexity of algebraic numbers. I, Ann. of Math., 165 (2007), 547–565.
B. Adamczewski, Y. Bugeaud and F. Luca, Sur la complexité des nombres algébriques, C. R. Acad. Sci., 339 (2004), 11–14.
J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations, Cambridge University Press (2003).
M. Drmota, C. Mauduit and J. Rivat, Normality along squares, J. Eur. Math. Soc, 21 (2019), 507–548.
S. Ferenczi and C. Mauduit, Transcendence of numbers with a low complexity expansion, J. Number Theory, 67 (1997), 146–161.
A. O. Gelfond, Sur les nombres qui ont des propriétés additives et multiplicatives données, Acta Arith., 13 (1968), 259–265.
K. Mahler, Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Math. Ann., 101 (1929), 342–362.
C. Mauduit and J. Rivat, La somme des chiffres des carrés, Acta Math., 203 (2009), 107–148.
E. Miyanohara, Remark on the transcendence of real numbers generated by Thue–Morse along squares, Notes Number Theory Discrete Math., 28 (2022), 159–166.
E. Miyanohara, An arthmetical property of the real numbers generated by Thue–Mose sequence along squares, Acta Math. Hungar, 170 (2023), 430–435.
Y. Moshe, On the subword complexity of Thue–Morse polynomial extractions, Theoret. Comput. Sci., 389 (2007), 318–329.
K. Nishioka, Mahler Functions and Transcendence, Lecture Notes in Math., vol. 1631, Springer (1996).
L. Spiegelhofer, Thue–Morse along the sequence of cubes, arXiv:2308.09498 (2023).
Acknowledgements
We thank Katsunobu Naruoka for valuable comments and warm encouragement. We also thank Professor Yohei Tachiya for informing me the details of the paper (arXiv:2312.06981). Finally, we thank the referee for careful reading and valuable comments. Especially, comments on the previous version of this paper were particularly helpful.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Miyanohara, E. Linear independence of the real numbers generated by the square and cube subsequences of Thue–Morse. Acta Math. Hungar. 172, 492–498 (2024). https://doi.org/10.1007/s10474-024-01417-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-024-01417-y