# A non-linear version of Bourgain’s projection theorem

### Pablo Shmerkin

University of British Columbia, Vancouver, Canada

## 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 $R_{2}$ or close to $3/2$ in $R_{3}$, then for $y∈A$ outside of a very sparse set, the pinned distance set ${∣x−y∣:x∈A}$ 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 $δ$-balls and $δ$-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.

## Cite this article

Pablo Shmerkin, A non-linear version of Bourgain’s projection theorem. J. Eur. Math. Soc. 25 (2023), no. 10, pp. 4155–4204

DOI 10.4171/JEMS/1283