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Irrationality and transcendence of continued fractions with algebraic integers (1902.04312v1)
Published 12 Feb 2019 in math.NT
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 $$ \limsup_{n \rightarrow \infty} \vert\alpha_n\vert{\frac{1}{Dd{n-1} \prod_{i=1}{n-2}(Ddi + 1)}} = \infty $$ implies that the continued fraction $\alpha = [0;\alpha_1, \alpha_2, \dots]$ is not an algebraic number of degree less than or equal to $D$.