On restricted families of projections in R^3

TL;DR

This paper investigates how the Hausdorff and packing dimensions of sets in three-dimensional space behave under restricted one-dimensional families of projections onto lines and planes, improving classical results and establishing new theorems for self-similar sets.

Contribution

It improves existing projection theorems by quantifying dimension preservation for sets with dimension greater than 1/2 and establishes a full Marstrand type theorem for self-similar sets without rotations.

Findings

For sets with Hausdorff dimension > 1/2, projections onto lines have packing dimension > 1/2 almost surely.

Projections onto planes preserve dimension > 1 almost surely.

Self-similar sets without rotations have positive length projections when dimension > 1.

Abstract

We study projections onto non-degenerate one-dimensional families of lines and planes in $\mathbb{R}^{3}$. Using the classical potential theoretic approach of R. Kaufman, one can show that the Hausdorff dimension of at most $1/2$-dimensional sets $B \subset \mathbb{R}^{3}$ is typically preserved under one-dimensional families of projections onto lines. We improve the result by an $\varepsilon$, proving that if $\dim_{\mathrm{H}} B = s > 1/2$, then the packing dimension of the projections is almost surely at least $\sigma(s) > 1/2$. For projections onto planes, we obtain a similar bound, with the threshold $1/2$ replaced by $1$. In the special case of self-similar sets $K \subset \mathbb{R}^{3}$ without rotations, we obtain a full Marstrand type projection theorem for one-parameter families of projections onto lines. The $\dim_{\mathrm{H}} K \leq 1$ case of the result follows from recent work of M. Hochman, but the $\dim_{\mathrm{H}} K > 1$ part is new: with this assumption, we prove that the projections have positive length almost surely.