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.

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