Integer-valued definable functions in ℝan,exp

Abstract

We give two variations on a result of Wilkie’s [A. J. Wilkie, Complex continuations of [Formula: see text]-definable unary functions with a diophantine application, J. Lond. Math. Soc. (2) 93(3) (2016) 547–566] on unary functions definable in [Formula: see text] that take integer values at positive integers. Provided that the function grows slower (in a suitable sense) than the function [Formula: see text], Wilkie showed that it must be eventually equal to a polynomial. Assuming a stronger growth condition, but only assuming that the function takes values sufficiently close to integers at positive integers, we show that the function must eventually be close to a polynomial. In a different variation we show that it suffices to assume that the function takes integer values on a sufficiently dense subset of the positive integers (for instance the primes), again under a stronger growth bound than that in Wilkie’s result.