A non-linear version of Bourgain’s projection theorem
Pablo Shmerkin
University of British Columbia, Vancouver, Canada
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Abstract
We prove a version of Bourgain’s projection theorem for parametrized families of 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 is a Borel set of Hausdorff dimension close to in or close to in , then for outside of a very sparse set, the pinned distance set has Hausdorff dimension at least , where is universal. Furthermore, the same holds if the distances are taken with respect to a 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 -balls and -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