For a real bivariate polynomial $f$ and finite set $A$ and $B$ of cardinality $n$, Elekes and Rónyai proved that either $f(A\times B)$ is much larger than $n$, or $f$ has a very specific form (essentially, $f(x,y)=x+y$). In the talk, I will tell about an analogue of this problem, where $A$ and $B$ are now Borel subsets of $\mathbb{R}$, each of Hausdorff dimension $\alpha$. In a recent work, joint with Josh Zahl, we prove that in this case $f(A\times B)$ will have Hausdorff dimension at least $\alpha+c$, where $c=c(\alpha)>0$, unless $f$ has the specific special form as above.
I will show a connection between the Elekes—Rónyai theorem just mentioned and the following result: Given a set $E$ in the plane, consider the set of exceptional points, for which the pinned distance set $\Delta_p(E)$ has small Hausdorff dimension, that is, close to ${\rm dim}(A)/2$. If this set has a positive dimension, then it must have a special structure. As a corollary one deduces a certain instance of the pinned Falconer’s distance problem.
For the proofs we apply a reduction to the discretized setting introduced by Katz and Tao. In the discretized setting, our proofs are inspired by their counterparts in the finite case, where the classical Szemerédi and Trotter incidence bound is replaced by a recent result of Shmerkin.
The talk is based on a joint work with Josh Zahl.