Irrationality of growth constants associated with polynomial recursions

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 $x_n \sim A α^{d^n}$, but little can be said about the constant $α$. In this paper, we show that $α$ 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 $G_n = A α^n + B + O(α^{-εn})$, where $A,B$ are algebraic and $α> 1$, and the sequence contains infinitely many integers, then $α$ is irrational or an integer.

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