Projection Theorems Using Effective Dimension

TL;DR

This paper applies computational complexity theory, specifically Kolmogorov complexity, to analyze fractal dimensions of projections in Euclidean spaces, extending classical results like Marstrand's theorem to broader contexts.

Contribution

It introduces two new results on Hausdorff and packing dimensions of projections using Kolmogorov complexity, and provides a novel proof of Marstrand's theorem based on computational theory.

Findings

Marstrand's theorem holds when Hausdorff and packing dimensions agree, even for non-analytic sets.

Provides a lower bound on the packing dimension of projections of arbitrary sets.

Offers a new proof of Marstrand's theorem using the theory of computing.

Abstract

In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrand's projection theorem, which shows that for every analytic set E, for almost every line L, the Hausdorff dimension of the orthogonal projection of E onto L is maximal. We use Kolmogorov complexity to give two new results on the Hausdorff and packing dimensions of orthogonal projections onto lines. The first shows that the conclusion of Marstrand's theorem holds whenever the Hausdorff and packing dimensions agree on the set E, even if E is not analytic. Our second result gives a lower bound on the packing dimension of projections of arbitrary sets. Finally, we give a new proof of Marstrand's theorem using the theory of computing.