The transcendence of growth constants associated with polynomial recursions

Abstract

Let [Formula: see text], [Formula: see text], be a polynomial of degree [Formula: see text]. Let [Formula: see text] be a sequence of integers satisfying [Formula: see text] Set [Formula: see text]. Then, under the assumption [Formula: see text], 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 [Formula: see text] is transcendental or [Formula: see text] can be an integer or a quadratic Pisot unit with [Formula: see text] being its conjugate over [Formula: see text]. In this paper, we study the nature of such [Formula: see text] without the assumption that [Formula: see text] is in [Formula: see text], and we prove that either the number [Formula: see text] is transcendental, or [Formula: see text] is a Pisot number with [Formula: see text] being the order of the torsion subgroup of the Galois closure of the number field [Formula: see text]. Other results presented in this paper investigate the solutions of the inequality [Formula: see text] in [Formula: see text], considering whether [Formula: see text] is rational or irrational. Here, [Formula: see text] represents a number field, and [Formula: see text]. The notation [Formula: see text] denotes the distance between [Formula: see text] and its nearest integer in [Formula: see text].