A survey on the Hausdorff dimension of intersections

TL;DR

This survey explores the Hausdorff dimension of intersections of Borel sets in Euclidean space under generic transformations, summarizing known results and open questions in geometric measure theory.

Contribution

It provides a comprehensive overview of existing results and open problems regarding the Hausdorff dimension of intersections under transformations.

Findings

Dimension of intersections often exceeds expected bounds

Results depend on the dimensions of the sets and the nature of transformations

Open problems remain in characterizing generic intersection dimensions

Abstract

Let $A$ and $B$ be Borel subsets of the Euclidean $n$-space with $\dim A + \dim B > n$. This is a survey on the question: what can we say about the Hausdorff dimension of the intersections $A\cap (g(B)+z)$ for generic orthogonal transformations $g$ and translations by $z$.