# Distance sets, orthogonal projections, and passing to weak tangents

## TL;DR

This paper explores the Assouad dimension analogues of key problems in geometric measure theory, demonstrating that sets with Assouad dimension >1 in the plane have distance sets with Assouad dimension 1 and extending projection theorems to higher dimensions.

## Contribution

It introduces new results on Assouad dimension for distance sets and projections, extending classical theorems and providing a framework for restricted projections.

## Key findings

- Sets with Assouad dimension >1 have distance sets with Assouad dimension 1 in the plane.
- Extended Fraser-Orponen projection theorem to higher dimensions.
- Provided estimates on the exceptional set of projections.

## Abstract

We consider the Assouad dimension analogues of two important problems in geometric measure theory. These problems are tied together by the common theme of `passing to weak tangents'. First, we solve an analogue of Falconer's distance set problem for Assouad dimension in the plane: if a planar set has Assouad dimension strictly greater than 1, then its distance set has Assouad dimension 1. We also obtain partial results in higher dimensions. Second, we consider how Assouad dimension behaves under orthogonal projection. We extend the planar projection theorem of Fraser and Orponen to higher dimensions, provide estimates on the (Hausdorff) dimension of the exceptional set of projections, and provide a recipe for obtaining results about restricted families of projections. We provide several illustrative examples throughout.