Integer‐valued definable functions

We present a dichotomy, in terms of growth at infinity, of analytic functions definable in the real exponential field which take integer values at natural number inputs. Using a result concerning the density of rational points on curves definable in this structure, we show that if a definable, analytic function f : [0, ∞) k →ℝ is such that f (ℕ k ) ⊆ ℤ, then either sup |x̄|⩽ r | f (x̄)| grows faster than exp( r δ ), for some δ>0, or f is a polynomial over ℚ.