Infinite products with algebraic numbers

TL;DR

This work establishes general criteria ensuring a nontrivial lower bound on the algebraic degree of numbers defined by infinite products involving algebraic integers. It develops a framework using a field tower, growth conditions, and divergence criteria to prove that $\deg_{\mathbb{K}}\left(\prod_{n=1}^{\infty}\left(1+\frac{b_n}{\alpha_n}\right)\right) > D$, and extends to two-dimensional products $\prod_{m=1}^{\infty}\left(1+\sum_{n=1}^{\infty}\frac{b_{n,m}}{\alpha_{n,m}}\right)$ under analogous hypotheses. The proofs synthesize height/ Mahler-measure techniques with Liouville-type lower bounds, bounding tails of the products and showing that a finite degree cannot approximate the infinite product, thereby forcing a higher degree. These results generalize earlier Erdős-type irrationality and degree bounds and suggest potential relaxations of the hypotheses, with connections to recent work on series and products involving algebraic numbers.

Abstract

We obtain general criteria for giving a lower bound on the degree of numbers of the form $\prod_{n=1}^\infty \left(1+\frac{b_n}{α_n}\right)$ or of the form $\prod_{m=1}^\infty \left(1+ \sum_{n=1}^\infty \frac{b_{n,m}}{α_{n,m}}\right)$, where the $α_n$ and $α_{n,m}$ are assumed to be algebraic integers, and the $b_n$ and $b_{n,m}$ are natural numbers. In each case, we give a lower bound of the degree over the smallest extension of $\mathbb{Q}$ containing all algebraic numbers in the expression. The criteria obtained depend on growth conditions on the involved quantities.

Theorems & Definitions (27)

• Theorem 1
• Theorem 2
• Remark
• Theorem 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9