The transcendence of growth constants associated with polynomial recursions

Let $P(x):=a_d{x}^d+\cdots +a_0\in \mathbb{Q}[x]$, $a_d>0$, be a polynomial of degree $d\ge 2$. Let $(x_n)$ be a sequence of integers satisfying $$x_{n+1}=P(x_n)\text{for all}n=0,1,2,\dots \text{and}{x}_n\to \infty \text{as}n\to \infty .$$

Set $\alpha :={lim}_{n\to \infty }{x}_n^{d^{-n}}$. Then, under the assumption $a_d^{1/(d-1)}\in \mathbb{Q}$, in a recent result by [A. Dubickas, Transcendency of some constants related to integer sequences of polynomial iterations, Ramanujan J.57 (2022) 569–581], 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 $\mathbb{Q}\left(\alpha ,a_d^{-\frac{1}{d-1}}\right)$.

Other results presented in this paper investigate the solutions of the inequality $\left|\right|q_1{\alpha }_1^n+\cdots +q_k{\alpha }_k^n+\beta \left|\right|<{𝜃}^n$ in $(n,q_1,\dots ,q_k)\in \mathbb{N}\times {(K^{\times })}^k$, considering whether $\beta$ is rational or irrational. Here, $K$ represents a number field, and $𝜃\in (0,1)$. The notation $\left|\right|x\left|\right|$ denotes the distance between $x$ and its nearest integer in $\mathbb{Z}$.