Irrationality of growth constants associated with polynomial recursions

Abstract

We consider integer sequences that satisfy a recursion of the form $x_{n+1} = P(x_n)$ for some polynomial $P$ of degree $d > 1$. If such a sequence tends to infinity, then it satisfies an asymptotic formula of the form $x_n \sim A \alpha^{d^n}$, but little can be said about the constant $\alpha$. In this paper, we show that $\alpha$ is always irrational or an integer. In fact, we prove a stronger statement: if a sequence $G_n$ satisfies an asymptotic formula of the form $G_n = A \alpha^n + B + O(\alpha^{-\epsilon n})$, where $A,B$ are algebraic and $\alpha > 1$, and the sequence contains infinitely many integers, then $\alpha$ is irrational or an integer.