Marstrand's Theorem Revisited: Projecting Sets of Dimension Zero

TL;DR

This paper refines Marstrand's projection theorem by extending it to Hausdorff dimension functions beyond power functions, including a logarithmic analogue, enhancing understanding of set projections.

Contribution

It introduces a refined version of Marstrand's theorem applicable to more delicate Hausdorff dimension functions, such as logarithmic dimensions.

Findings

Established a projection theorem for logarithmic Hausdorff dimension.

Extended classical results to finer dimension functions.

Provided new tools for analyzing projections of fractal sets.

Abstract

We establish a refinement of Marstrand's projection theorem for Hausdorff dimension functions finer than the usual power functions, including an analogue of Marstrand's Theorem for logarithmic Hausdorff dimension.