Model Theory Seminar
James Freitag
UIC
Diophantine properties of sets definable in o-minimal structures
Abstract: In any o-minimal expansion of the real numbers, the number of integer points in one-dimensional sets defined analytically (or in any non-semi-algebraic manner) is more tightly controlled than one might naively expect.
In particular, the number of integer points in a ball of radius $R$ grows asymptotically slower than $R^ \epsilon$ for any $\epsilon >0$. In certain o-minimal expansions, one can do even better. We will also discuss applications of these results and limitations.
Results of this form can be viewed as precursors to a much more general result, the Pila-Wilkie Theorem, which the seminar will move onto next.
Wednesday February 15, 2017 at 10:00 AM in SEO 427