  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.
