Abstract
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(ℕk) ⊆ ℤ, then either sup|x̄|⩽ r |f(x̄)| grows faster than exp(rδ), for some δ>0, or f is a polynomial over ℚ.
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