Let X ⊂ R n be a set that is definable in an o-minimal structure over R. This article shows that in a suitable sense, there are very few rational points of X which do not lie on some connected semialgebraic subset of X of positive dimension.
IntroductionThis article is concerned with the distribution of rational and integer points on certain nonalgebraic sets in R n . To contextualize the kind of results sought and, in particular, to motivate the present setting of definable sets in o-minimal structures over R (see Definition 1.7), we begin by describing earlier results.The ideas pursued here grew from the article [4] of Bombieri and Pila, where a technique using elementary real-variable methods and elementary algebraic geometry was used to establish upper bounds for the number of integer points on the graphs of functions y = f (x) under various natural smoothness and convexity hypotheses. Results were obtained for f variously assumed to be (sufficiently) smooth, algebraic, or real analytic. Several results concerned the homothetic dilation of a fixed graph X : y = f (x).For a real number t ≥ 1 (which is always tacitly assumed), the homothetic dilation of X by t is the set tX = { tx 1 , . . . , tx n : x 1 , . . . , x n ∈ X}. By X(Z) we denote the subset of X comprising the points with integer coordinates.Suppose now that X is the graph of a function f : [0, 1] → R. Trivially, one has #(tX)(Z) ≤ t + 1 (with equality, e.g., for f (x) = x and positive integral t). According to Jarník [15], a strictly convex arc : y = g(x) of length contains at most 3(4π) −1/3 2/3 + O( 1/3 ) integer points. (And, moreover, the exponent and constant are the best possible.) So if X is strictly convex, one infers that #(tX)(Z) ≤ c(X)t 2/3 . PILA and WILKIE However, showed that a substantially better estimate may be obtained if f is assumed to be C 3 and strictly convex; namely, #(tX)(Z) ≤ c(X, )t 3/5+ for all positive . (Regarding this circle of limited smoothness problems, also see [30], [21].)Counting integer points on tX is of course the same as counting points m/t, n/t on X, and in this guise such questions arose in work of Sarnak [29] on Betti numbers of abelian covers. He conjectured that if f is C ∞ and strictly convex, then in fact,for all positive . The exponent 1/2 is the best possible here in view of f (x) = x 2 . This conjecture was the starting point of [4], where it is affirmed.If f is assumed to be transcendental analytic, however, then the exponent may be reduced to , and this result of [4] is the prototype for the results presented here. THEOREM 1.2 ([4, Theorem 1]) Let f : [0, 1] → R be a transcendental real-analytic function. Let X be the graph of f , and let > 0. There is a constant c(X, ) such that PILA and WILKIE
