Algebraic degree of series of reciprocal algebraic integers

TL;DR

This paper establishes conditions under which linear combinations of certain infinite series of reciprocals of algebraic integers have algebraic degree exceeding any fixed integer, based on growth conditions of the algebraic integers involved.

Contribution

It provides new sufficient conditions ensuring that such linear combinations have arbitrarily high algebraic degree, extending understanding of algebraic degrees of series of reciprocals.

Findings

Linear combinations can have arbitrarily high algebraic degree under specified conditions.

Rapid growth of algebraic integers' moduli influences the algebraic degree of series.

Conditions on coefficients ensure high algebraic degree of the sums.

Abstract

In this paper, I give sufficient conditions for any linear combination in $\mathbb{Q}$ of numbers $\sum_{n=1}^{\infty}\frac{b_{1,n}}{\alpha_{1,n}}$, $\ldots$, $\sum_{n=1}^{\infty}\frac{b_{K,n}}{\alpha_{K,n}}$ 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.