Quantitative control on the Carleson $\epsilon$-function determines regularity

Carleson’s ${\epsilon }^2$-conjecture states that for Jordan domains in ${\mathbb{R}}^2$, points on the boundary where tangents exist can be characterized in terms of the behavior of the $\epsilon$-function. This conjecture, which was fully resolved by Jaye, Tolsa, and Villa in 2021, established that qualitative control on the rate of decay of the Carleson $\epsilon$-function implies the existence of tangents, up to a set of measure zero. We prove that quantitative control on the rate of decay of this function gives quantitative information on the regularity of the boundary.