A transcendence criterion for infinite products

Abstract

We prove a transcendence criterion for certain infinite products of algebraic numbers. Namely, for an increasing sequence of positive integers $a_n$ and an algebraic number $\alpha >1$, we consider the convergent infinite product ${\prod }_n([{\alpha }^{a_n}]/{\alpha }^{a_n})$, where $[\cdot ]$ stands for the integral part. We prove (Thm. 1) that its value is transcendental, under certain hypothesis; Thm. 3 will show that such hypothesis are in a sense unavoidable.