A non-linear version of Bourgain’s projection theorem

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