Diophantine properties of sets definable in o-minimal structures

Publisher： Cambridge University Press

E-ISSN： 1943-5886|69|3|851-861

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.69, Iss.3, 2004-09, pp. : 851-861

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Abstract

Let be an o-minimal expansion of the ordered field of real numbers , and let S be an -definable subset (parameters allowed unless otherwise stated) of ℝ n . In this note I investigate questions concerning the distribution of points on S with integer coordinates. My main theorem gives an estimate which, though probably far from best possible, at least shows that the o-minimal assumption does have diophantine consequences. This is, perhaps, surprising in view of the flexibility that we now seem to have in constructing o-minimal expansions of (see, e. g. [7], [8], [9]).

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