A transcendence criterion for infinite products
The authors prove a transcendence criterion for certain infinite products of algebraic numbers. For an increasing sequence of positive integers \(a_n\) and an algebraic number \(\alpha>1\), they study the convergent infinite product \(\prod_{n}([\alpha^{a_n}]/\alpha^{a_n})\), where \([\cdot]\) denotes the integral part. It is proved in Theorem 1 that its value is transcendental, under certain hypotheses; whereas Theorem 3 shows that such hypotheses are in a sense unavoidable. This is done by applying a result of the first author and \textit{U. Zannier} [Acta Math. 193, No. 2, 175--191 (2004; Zbl 1175.11036)].
scientific article; zbMATH DE number 5292681