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

Carleson's $\varepsilon^2$-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 $\varepsilon$-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 $\varepsilon$-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.