Seventy Years of Fractal Projections

TL;DR

This paper reviews the historical development and significance of Marstrand's projection theorems in fractal geometry, highlighting their foundational role and ongoing influence in the field over the past 70 years.

Contribution

It provides a comprehensive overview of the evolution, variants, and applications of Marstrand's projection theorems in fractal geometry.

Findings

Marstrand's theorems relate Hausdorff dimension to projections

The theorems have inspired numerous variants and applications

They remain central to ongoing research in fractal geometry

Abstract

Seventy years ago, John Marstrand published a paper which, among other things, relates the Hausdorff dimension of a plane set to the dimensions of its orthogonal projections onto lines. For some time this paper attracted little attention, but over the past 40 years Marstrand's projection theorems have become the prototype for many results in fractal geometry with numerous variants and applications and they continue to motivate leading research.