Assume that Q is a relation on R^s of arity s definable in an o-minimal expansion of R. I will discuss how certain extremal asymptotic behaviors of the sizes of the intersections of Q with finite n × … × n grids, for growing n, can only occur if Q is closely connected to a certain algebraic structure.
On the one hand, if the projection of Q onto any s-1 coordinates is finite-to-one but Q has maximal size intersections with some grids (of size >n^(s-1 – ε)), then Q restricted to some open set is, up to coordinatewise homeomorphisms, of the form x_1+…+x_s=0. This is a special case of the recent generalization of the Elekes-Szabó theorem to any arity and dimension in which general abelian Lie groups arise (joint work with Kobi Peterzil and Sergei Starchenko).On the other hand, if Q omits a finite complete s-partite hypergraph but can intersect finite grids in more that than n^(s-1 + ε) points, then the real field can be definably recovered from Q (joint work with Abdul Basit, Sergei Starchenko, Terence Tao and Chieu-Minh Tran).
I will explain how these results are connected to the model-theoretic trichotomy principle and discuss variants for higher dimensions, and for stable structures with distal expansions.
- Séminaire Géométrie et théorie des modèles