Remark on the transcendence of real number generated by Thue–Morse along squares

Eiji Miyanohara
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 2, Pages 159–166
DOI: 10.7546/nntdm.2022.28.2.159-166
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Eiji Miyanohara
Tokyo, Japan

Abstract

In 1929, Mahler proved that the real number generated by Thue–Morse sequence is transcendental. Later, Adamczewski and Bugeaud gave a different proof of the transcendence of this number using a combinatorial transcendence criterion. Moreover, Kumar and Meher gave the generalization of the combinatorial transcendence criterion under the subspace Lang conjecture. In this paper, we prove under the subspace Lang conjecture that the real number generated by Thue–Morse along squares is transcendental by using the combinatorial transcendence criterion of Kumar and Meher.

Keywords

• Thue–Morse along squares
• Transcendence

2020 Mathematics Subject Classification

• 11B99
• 11J91

References

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Manuscript history

• Received: 25 May 2021
• Revised: 21 March 2022
• Accepted: 2 April 2022
• Online First: 5 April 2022