Quantitative control of the epsilon-function determines regularity

A longstanding conjecture of Carleson stated that the tangent points of the boundaries of certain planar domains can be characterized by the behavior of the Carleson -function. This conjecture, which was fully resolved by Jaye, Tolsa, and Villa in 2021, established that having some Dini type control of the Carleson -function implied the existence of tangents. A natural question is whether quantitative control on this function implies better regularity results. In this talk, we will present results that give a positive answer to this question. This is ongoing work.