An o-minimal version of Gromov's "Algebraic Reparameterization Lemma" with a diophantine application
Seminar Room 1, Newton Institute
Suppose that M is any o-minimal expansion of a real closed field and that X is any definable subset of M^n which contains no infinite semi-algebraic subset of M^n. Then for every real number s>0, X contains at most O(H^s) points with rational co-ordinates of height at most H (for all H>0, where the implied constant depends only on X and s). The diophantine part of the proof follows a method of Bombieri and Pila which was subsequently modified by Pila to establish the result in the case when X is a subanalytic subset of real n-space. However, it seems to me that the analytic part of Pila's proof can be considerably simplified by working in a general o-minimal setting and invoking uniformity in parameters. This is applied to a generalization of a result of Gromov, namely that for any given r, any bounded definable set may be represented as a finite union of images of open unit cubes (of various dimensions) by C^r functions all of whose derivatives (up to the r'th) are bounded by one.
E Bombieri and J Pila, 'The number of integral points on arcs and ovals.' Duke Math J., Vol 59(2),Oct 1989,337-357.
M Gromov, 'Entropy, homology and semialgebraic geometry [after Y. Yomdin].' Sem Bourbaki 663, Asterisque 145-146 (1987) 225-240.
J Pila, 'Rational points of a subanalytic set'. Preprint, Nov 2004.(I'll bring this with me.)