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arXiv · 1902.04312

Irrationality and transcendence of continued fractions with algebraic integers

Simon Bruno Andersen·Simon Kristensen
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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 $D$.

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Simon Bruno Andersen, Simon Kristensen. 2019-02-12. Irrationality and transcendence of continued fractions with algebraic integers. https://arxiv.org/abs/1902.04312

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