A nonlinear projection theorem for Assouad dimension and applications

TL;DR

This paper establishes a nonlinear projection theorem for Assouad dimension and applies it to solve the planar distance set problem, analyze radial projections, and explore sum-product phenomena, providing sharp estimates and resolving critical cases.

Contribution

It introduces a general nonlinear projection theorem for Assouad dimension and applies it to several problems, including the planar distance set problem and radial projections, with sharp estimates and complete solutions.

Findings

Resolved the planar distance set problem for Assouad dimension.

Established a radial projection theorem with sharp estimates.

Connected the higher-dimensional problem to orthogonal projection exceptions.

Abstract

We prove a general nonlinear projection theorem for Assouad dimension. This theorem has several applications including to distance sets, radial projections, and sum-product phenomena. In the setting of distance sets we are able to completely resolve the planar distance set problem for Assouad dimension, both dealing with the awkward `critical case' and providing sharp estimates for sets with Assouad dimension less than 1. In the higher dimensional setting we connect the problem to the dimension of the set of exceptions in a related (orthogonal) projection theorem. We also obtain results on pinned distance sets and our results still hold when the distances are taken with respect to a sufficiently curved norm. As another application we prove a radial projection theorem for Assouad dimension with sharp estimates on the Hausdorff dimension of the exceptional set.