ALGEBRAIC DEGREE OF SERIES OF RECIPROCAL ALGEBRAIC INTEGERS

Abstract

We give sufficient conditions for any linear combination over $\mathbb{Q}$ of numbers ${\sum }_{n=1}^{\infty }{b}_{1,n}∕{\alpha }_{1,n},\dots ,{\sum }_{n=1}^{\infty }{b}_{K,n}∕{\alpha }_{K,n}$ to have algebraic degree greater than an arbitrary fixed integer $D$ when the numbers ${\alpha }_{i,n}$ are algebraic integers of sufficiently rapidly increasing modulus and the $b_{i,n}$ are positive integers that are not too large.