# On the Assouad dimension of projections

## TL;DR

This paper proves a strong projection theorem for Assouad dimension, showing that for most directions, the Assouad dimension of the projection of a set in the plane is at least the minimum of the set's Assouad dimension and 1, highlighting a unique property not shared by other fractal dimensions.

## Contribution

The paper establishes a Marstrand-type projection theorem for Assouad dimension, demonstrating that the Assouad dimension of projections is generally preserved outside a negligible set of directions.

## Key findings

- Assouad dimension of projections is at least min{original dimension, 1} for almost all directions.
- The result holds outside a set of directions with Hausdorff dimension zero.
- This property is specific to Assouad dimension and does not hold for Hausdorff or packing dimensions.

## Abstract

Let $F \subset \mathbb{R}^{2}$, and let $\dim_{\mathrm{A}}$ stand for Assouad dimension. I prove that $\dim_{\mathrm{A}} \pi_{e}(F) \geq \min\{\dim_{\mathrm{A}} F,1\}$ for all $e \in S^{1}$ outside of a set of Hausdorff dimension zero. This is a strong variant of Marstrand's projection theorem for Assouad dimension, whose analogue is not true for other common notions of fractal dimension, such as Hausdorff or packing dimension.