Algebraic degree of series of reciprocal algebraic integers
Mathias Løkkegaard Laursen
TL;DR
This work provides sufficient conditions under which any nontrivial $\mathbb{Q}$-linear combination of the infinite sums $\sum_{n=1}^{\infty} \frac{b_{i,n}}{\alpha_{i,n}}$ has algebraic degree strictly greater than a fixed integer $D$, with $\alpha_{i,n}$ being algebraic integers of rapidly increasing modulus and $b_{i,n}$ small positive integers. The approach combines height/measure methods (Weil height $H$, Mahler measure $M$, and house) with Liouville-type lower bounds and a carefully controlled growth framework, including an open half-plane constraint via $\Re_\zeta$, to guarantee non-conjugacy of partial sums. The main result generalizes prior work on reciprocal algebraic-integer series by Hančl, Hančl–Nair, and Andersen–Kristensen, and yields that $\deg\gamma > D$ for any nontrivial linear combination $\gamma$ of the sums. In the case $K=1$, a streamlined corollary obtains large algebraic degree for $\sum_{n} 1/\alpha_n$ under similar growth conditions, highlighting implications for algebraic independence and Diophantine approximation of reciprocal sums of algebraic integers.
Abstract
In this paper, I give sufficient conditions for any linear combination in $\mathbb{Q}$ of numbers $\sum_{n=1}^{\infty}\frac{b_{1,n}}{α_{1,n}}$, $\ldots$, $\sum_{n=1}^{\infty}\frac{b_{K,n}}{α_{K,n}}$ to have algebraic degree greater than an arbitrary fixed integer $D$ when the numbers $α_{i,n}$ are algebraic integers of sufficiently rapidly increasing modulus and the $b_{i,n}$ are positive integers that are not too large.