On the transcendence of growth constants associated with polynomial recursions

TL;DR

This paper investigates the transcendence and algebraic properties of growth constants associated with polynomial recursions, extending previous results by removing certain rationality assumptions and exploring solutions to related Diophantine inequalities.

Contribution

It generalizes earlier work by analyzing the nature of the limit constant without the rationality constraint on the polynomial's leading coefficient.

Findings

Either the growth constant is transcendental or its power is a Pisot number.

Characterization of the algebraic nature of the limit constant without rationality assumptions.

Results on solutions to inequalities involving exponential sums in number fields.

Abstract

Let $P(x):=a_d x^d+\cdots+a_0\in\mathbb{Q}[x]$, $a_d>0$, be a polynomial of degree $d\geq 2$. Let $(x_n)$ be a sequence of integers satisfying \begin{equation*} x_{n+1}=P(x_n)\mbox{for all}\quad n=0,1,2\ldots,\quad\mbox{and} \quad x_n\to\infty\quad\mbox{as}\quad n\to\infty. \end{equation*} Set $\alpha:=\lim_{n\to\infty} x^{d^{-n}}_n$. Then, under the assumption $a_d^{1/(d-1)}\in\mathbb{Q}$, in a recent result by Dubickas \cite{dubickas}, either $\alpha$ is transcendental, or $\alpha$ can be an integer, or a quadratic Pisot unit with $\alpha^{-1}$ being its conjugate over $\mathbb{Q}$. In this paper, we study the nature of such $\alpha$ without the assumption that $a_d^{1/(d-1)}$ is in $\mathbb{Q}$, and we prove that either the number $\alpha$ is transcendental, or $\alpha^h$ is a Pisot number with $h$ being the order of the torsion subgroup of the Galois closure of the number field $\mathbb{Q}(\alpha, a_d^{-\frac{1}{d-1}})$. Other results presented in this paper investigate the solutions of the inequality $||q_1 \alpha_1^n+\cdots+q_k \alpha_k^n +\beta||<\theta^n$ in $(n,q_1,\ldots,q_k)\in \mathbb{N}\times(K^\times)^k$, considering whether $\beta$ is rational or irrational. Here, $K$ represents a number field, and $\theta\in (0,1)$. The notation $||x||$ denotes the distance between $x$ and its nearest integer in $\mathbb{Z}$.