The transcendence of growth constants associated with polynomial recursions

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.