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

TL;DR

This paper demonstrates the existence of uncountably many transcendental numbers generated by digital pattern sequences, extending previous results that only identified countably many such numbers.

Contribution

It introduces the concept of generalized k-regular sequences and applies combinatorial transcendence criteria to prove the uncountability of these transcendental numbers.

Findings

Uncountably many transcendental numbers can be generated by digital pattern sequences.

Generalized k-regular sequences are key to establishing transcendence.

The method extends previous countability results to uncountability.

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.