Assouad dimension influences the box and packing dimensions of orthogonal projections

TL;DR

This paper explores how the Assouad dimension and related concepts affect the behavior of box and packing dimensions under orthogonal projections, establishing sharp thresholds and bounds for typical projections.

Contribution

It demonstrates that the (quasi-)Assouad dimension bounds the preservation of box and packing dimensions under projections and provides sharp thresholds and bounds for the exceptional set.

Findings

Assouad dimension bounds the projection behavior of sets.

The threshold for dimension preservation is sharp.

Bounds on the exceptional set of projections are established.

Abstract

We present several applications of the Assouad dimension, and the related quasi-Assouad dimension and Assouad spectrum, to the box and packing dimensions of orthogonal projections of sets. For example, we show that if the (quasi-)Assouad dimension of $F \subseteq \rn$ is no greater than $m$, then the box and packing dimensions of $F$ are preserved under orthogonal projections onto almost all $m$-dimensional subspaces. We also show that the threshold $m$ for the (quasi-)Assouad dimension is sharp, and bound the dimension of the exceptional set of projections strictly away from the dimension of the Grassmannian.