Exeptional set estimates for the Hausdorff dimension of intersections

TL;DR

This paper investigates the size of the set of rotations and translations for which the intersection of two Borel sets in Euclidean space has a Hausdorff dimension smaller than the typical expected value, providing bounds on this exceptional set.

Contribution

The paper derives estimates for the exceptional set of rotations and translations where the Hausdorff dimension of intersections falls below the generic lower bound.

Findings

Estimates for the measure of the exceptional set of rotations and translations.

Conditions under which the Hausdorff dimension of intersections meets the expected lower bound.

Quantitative bounds on the size of the exceptional set.

Abstract

For Borel subsets A and B of the Euclidean n-space the intersection of A with generic rotations and translations of B has often Hausdorff dimension at least dim A + dim B - n. Estimates for the exceptional set of rotations are derived.