A Baire Category Approach to Besicovitch's Theorem and Measure Regularity

Summary · qwen2.5:32b

The article demonstrates that Besicovitch's Theorem can be proven within the subsystem $\mathsf{ACA}_0$ by reformulating its proof using a Baire Category argument, and shows that the witnessing subset is computable from one jump of the original set. This equivalence between the Baire Category Theorem for Closed Sets ($\mathsf{BCTC}$) and $\mathsf{ACA}_0$ provides new insights into measure regularity properties and their computational complexity.

Excerpt

arXiv:2602.00940v2 Announce Type: replace Abstract: By reformulating the classical proof as a Baire Category argument, we show that Besicovitch's Theorem on the existence of subsets of finite Hausdorff measure is provable in $\mathsf{ACA}_0$, and additionally that the witnessing subset is computable from one jump of the original set. We show that the corresponding formulation of Baire Category, which we call Baire Category Theorem for Closed Sets ($\mathsf{BCTC}$), is equivalent to $\mathsf{ACA}_0$, contrasting with previous results on the reverse math strength of Baire Category variants. We also examine the implications of $\mathsf{BCTC}$ for a class of monotone functions on closed sets, and explore how changing the representation of a closed set affects the reverse math strength of its measure regularity properties.