Irrationality and transcendence of continued fractions with algebraic integers

TL;DR

This paper generalizes previous results on the irrationality and transcendence of continued fractions with algebraic integer entries, establishing conditions under which such continued fractions are transcendental or irrational.

Contribution

It extends earlier work by providing new growth conditions on algebraic integer sequences that guarantee the transcendence or irrationality of the resulting continued fractions.

Findings

Certain growth conditions imply the continued fraction is transcendental.

The results apply to sequences of algebraic integers with bounded degree.

The work generalizes previous irrationality and transcendence criteria.

Abstract

We extend a result of Han\v{c}l, Kolouch and Nair on the irrationality and transcendence of continued fractions. We show that for a sequence $\{\alpha_n\}$ of algebraic integers of bounded degree, each attaining the maximum absolute value among their conjugates and satisfying certain growth conditions, the condition $$ \limsup_{n \rightarrow \infty} \vert\alpha_n\vert^{\frac{1}{Dd^{n-1} \prod_{i=1}^{n-2}(Dd^i + 1)}} = \infty $$ implies that the continued fraction $\alpha = [0;\alpha_1, \alpha_2, \dots]$ is not an algebraic number of degree less than or equal to $D$.