Digital pattern and transcendence via generalized $k$-regular sequences

Abstract

In this paper, we prove that there are uncountable many real transcendental numbers, which are generated by digital pattern sequences. This generalizes the main theorem in Morton and Mourant, which states the existence of countable many similar numbers. Our method relies on the combinatorial quantitative transcendence criterion established by Adamczewski and Bugeaud and properties of generalized k-regular sequences, which is introduced by this paper.