The Assouad dimensions of projections of planar sets

TL;DR

This paper studies the Assouad dimensions of projections of planar sets, revealing that for general sets the dimension of projections is almost surely at least the minimum of 1 and the original dimension, with detailed results for self-similar sets.

Contribution

It provides new results on the Assouad dimensions of projections, including a dichotomy for self-similar sets based on rotation groups and shows the failure of a Falconer-type theorem for Assouad dimension.

Findings

Projections of sets with Assouad dimension s have dimension at least min{1,s} almost surely.

For self-similar sets, the Assouad dimension of projections can be either s or 1 depending on the measure.

Assouad dimension is not almost surely constant under random translation vectors for self-similar sets.

Abstract

We consider the Assouad dimensions of orthogonal projections of planar sets onto lines. Our investigation covers both general and self-similar sets. For general sets, the main result is the following: if a set in the plane has Assouad dimension $s \in [0,2]$, then the projections have Assouad dimension at least $\min\{1,s\}$ almost surely. Compared to the famous analogue for Hausdorff dimension -- namely \emph{Marstrand's Projection Theorem} -- a striking difference is that the words `at least' cannot be dispensed with: in fact, for many planar self-similar sets of dimension $s < 1$, we prove that the Assouad dimension of projections can attain both values $s$ and $1$ for a set of directions of positive measure. For self-similar sets, our investigation splits naturally into two cases: when the group of rotations is discrete, and when it is dense. In the `discrete rotations' case we prove the following dichotomy for any given projection: either the Hausdorff measure is positive in the Hausdorff dimension, in which case the Hausdorff and Assouad dimensions coincide; or the Hausdorff measure is zero in the Hausdorff dimension, in which case the Assouad dimension is equal to 1. In the `dense rotations' case we prove that every projection has Assouad dimension equal to one, assuming that the planar set is not a singleton. As another application of our results, we show that there is no \emph{Falconer's Theorem} for Assouad dimension. More precisely, the Assouad dimension of a self-similar (or self-affine) set is not in general almost surely constant when one randomises the translation vectors.