A non-linear version of Bourgain’s projection theorem

Abstract

We prove a version of Bourgain’s projection theorem for parametrized families of $C^2$ maps, which refines the original statement even in the linear case by requiring non-concentration only at a single natural scale. As one application, we show that if $A$ is a Borel set of Hausdorff dimension close to $1$ in ${\mathbb{R}}^2$ or close to $3/2$ in ${\mathbb{R}}^3$, then for $y\in A$ outside of a very sparse set, the pinned distance set $\{|x-y|:x\in A\}$ has Hausdorff dimension at least $1/2+c$, where $c$ is universal. Furthermore, the same holds if the distances are taken with respect to a $C^2$ norm of positive Gaussian curvature. As further applications, we obtain new bounds on the dimensions of spherical projections, and an improvement over the trivial estimate for incidences between $\delta$-balls and $\delta$-neighborhoods of curves in the plane, under fairly general assumptions. The proofs depend on a new multiscale decomposition of measures into “Frostman pieces” that may be of independent interest.