On the dimension of exceptional parameters for nonlinear projections, and the discretized Elekes-R\'onyai theorem

We consider four related problems. (1) Obtaining dimension estimates for the set of exceptional vantage points for the pinned Falconer distance problem. (2) Nonlinear projection theorems, in the spirit of Kaufman, Bourgain, and Shmerkin. (3) The parallelizability of planar $d$-webs. (4) The Elekes-R\'onyai theorem on expanding polynomials. Given a Borel set $A$ in the plane, we study the set of exceptional vantage points, for which the pinned distance $\Delta_p(A)$ has small dimension, that is, close to $(\dim A)/2$. We show that if this set has positive dimension, then it must have very special structure. This result follows from a more general single-scale nonlinear projection theorem, which says that if $\phi_1,\phi_2,\phi_3$ are three smooth functions whose associated 3-web has non-vanishing Blaschke curvature, and if $A$ is a $(\delta,\alpha)_2$-set in the sense of Katz and Tao, then at least one of the images $\phi_i(A)$ must have measure much larger than $|A|^{1/2}$, where $|A|$ stands for the measure of $A$. We prove analogous results for $d$ smooth functions $\phi_1,\ldots,\phi_d$, whose associated $d$-web is not parallelizable. We use similar tools to characterize when bivariate real analytic functions are "dimension expanding" when applied to a Cartesian product: if $P$ is a bivariate real analytic function, then $P$ is either locally of the form $h(a(x) + b(y))$, or $P(A,B)$ has dimension at least $\alpha+c$ whenever $A$ and $B$ are Borel sets with Hausdorff dimension $\alpha$. Again, this follows from a single-scale estimate, which is an analogue of the Elekes-R\'onyai theorem in the setting of the Katz-Tao discretized ring conjecture.