A Baire Category Approach to Besicovitch's Theorem

Abstract: One of the fundamental results from geometric measure theory is Besicovitch's theorem from 1952, which states that any closed subset of Euclidean space having infinite Hausdorff measure contains a compact subset with positive finite Hausdorff measure. However, the computatibility theoretic and reverse mathematical complexity of this result have not been extensively studied. In this talk, we will introduce a variant of the Baire Category Theorem, and show how we can reframe Besicovitch's original proof through that lens. This approach not only confirms that the theorem is provable in $\text{ACA}_0$, but demonstrates how a witnessing subset can be computed from just one jump of the original set.

differential geometrylogicmetric geometry

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic