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Transcendence criteria for infinite products of algebraic numbers

Mathias L. Laursen

TL;DR

This work develops new transcendence criteria for rapidly converging infinite products of algebraic numbers by applying Schmidt's Subspace Theorem. It introduces and leverages $\Sigma$-irrationality and $(\Pi,K)$-irrationality to obtain both irrationality and transcendence criteria for infinite products, with sharper results when the base field has degree $d>1$. The main contributions include rational and algebraic (number-field) variants, allowing a finite $A$ in the transcendence criterion, and broad applicability to explicit constructions (e.g., Fibonacci-type sequences and algebraic units). The results generalize Erdős-type irrationality theorems and provide practical tools for proving transcendence of carefully chosen infinite products and related series.

Abstract

Using an application of Schmidt's Subspace Theorem, this paper gives new transcendence criteria for rapidly converging infinite products of algebraic numbers. The paper also improves existing criteria for irrationality of products and criteria for irrationality and transcendence of infinite series. These results generalize a classical theorem on the irrationality of infinite series due to Erdős.

Transcendence criteria for infinite products of algebraic numbers

TL;DR

This work develops new transcendence criteria for rapidly converging infinite products of algebraic numbers by applying Schmidt's Subspace Theorem. It introduces and leverages -irrationality and -irrationality to obtain both irrationality and transcendence criteria for infinite products, with sharper results when the base field has degree . The main contributions include rational and algebraic (number-field) variants, allowing a finite in the transcendence criterion, and broad applicability to explicit constructions (e.g., Fibonacci-type sequences and algebraic units). The results generalize Erdős-type irrationality theorems and provide practical tools for proving transcendence of carefully chosen infinite products and related series.

Abstract

Using an application of Schmidt's Subspace Theorem, this paper gives new transcendence criteria for rapidly converging infinite products of algebraic numbers. The paper also improves existing criteria for irrationality of products and criteria for irrationality and transcendence of infinite series. These results generalize a classical theorem on the irrationality of infinite series due to Erdős.

Paper Structure

This paper contains 5 sections, 17 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications
  4. Preliminaries
  5. Proof of Main Theorems

Key Result

Theorem 1.1

Let $\varepsilon>0$, and let $\{a_n\}_{n=1}^\infty$ be an increasing sequence of integers such that $a_n\ge n^{1+\varepsilon}$ for all $n$. Suppose Then the sequence $\{a_n\}_{n=1}^\infty$ is $\Sigma$-irrational.

Theorems & Definitions (32)

  • Theorem 1.1: Erdős
  • Theorem 1.2: Kristensen and Laursen
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • ...and 22 more