Assume that $Q$ is a relation on $\mathbb{R}^s$ of arity $s$ definable in an $o$-minimal expansion of $\mathbb{R}$. I will discuss how certain extremal asymptotic behaviors of the sizes of the intersections of $Q$ with finite $n \times \ldots \times n$ 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 $>n^{s-1 - \varepsilon}$), then $Q$ restricted to some open set is, up to coordinatewise analytic bijections, of the form $x_1+ \ldots +x_s=0$. 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, from 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 than $n^{s-1 + \varepsilon}$ 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.