Generalized $k$-regular sequences II:digital pattern and transcendence

TL;DR

This paper proves that uncountably many real numbers generated by digital pattern sequences are transcendental, extending previous results that only applied to countably many, using combinatorial transcendence criteria and properties of generalized k-regular sequences.

Contribution

It generalizes the known result from countable to uncountable sets of real numbers generated by digital pattern sequences, utilizing advanced combinatorial and sequence properties.

Findings

Uncountably many digital pattern sequence-generated numbers are transcendental.

Extension of previous countable results to uncountable sets.

Application of combinatorial transcendence criteria to generalized k-regular sequences.

Abstract

In this paper, we prove that an uncountable quantity of real numbers generated by digital pattern sequences gives the transcendental number. This result gives a generalization of Main theorem in Morton and Mourant [MortM], which state that countable real numbers generated by digital pattern sequences gives the transcendental number. Our method relies on the combinatorial quantitative transcendence criterion established by Adamczewski-Bugeaud [AdB2] and the properties of generalized $k$-regular sequences, which is introduced by the author [Mi2].