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

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.