Transcendence of values of the iterated exponential function at algebraic points

TL;DR

This paper investigates the behavior of iterated exponential functions at algebraic points, revealing a relationship between the convergence of such sequences to algebraic numbers and the algebraic order of the base.

Contribution

It establishes an approximation formula for the number of algebraic numbers with a given order whose iterated exponentials converge to algebraic numbers.

Findings

Number of algebraic bases with order k converging to algebraic numbers approximates (e-1/e)φ(k).

Introduces a link between algebraic order and convergence behavior of iterated exponentials.

Provides a new perspective on transcendence and algebraic properties of iterated exponential sequences.

Abstract

We say that the order of an algebraic number $A$ is the minimum of positive integers $k$ such that $A^k$ is rational. In this paper, we show that the number of algebraic numbers $A$ with order $k$ such that \[ A,\ A^A,\ A^{A^A},\ \ldots \] converges to an algebraic number is approximated by $(e-1/e)\varphi (k)$. Here $\varphi(k)$ denotes Euler's totient function.