doi:10.1112/blms/bds059 Integer-valued definable functions

O. Jones, M. E. M. Thomas and A. J. Wilkie 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. 1