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) similar to A alpha(dn), but little can be said about the constant alpha. In this paper, we show that ff is always irrational or an integer. In fact, we prove a stronger statement: if a sequence (G(n))(n >= 0) 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 ff is irrational or an integer.