On the projections of almost Ahlfors regular sets

TL;DR

This paper demonstrates the sharpness of the Kaufman projection theorem for almost Ahlfors regular sets, contrasting with recent improvements for strictly Ahlfors regular sets, thereby clarifying the theorem's limits.

Contribution

It establishes the sharpness of the Kaufman projection theorem within the class of almost Ahlfors regular sets, extending understanding of projection properties.

Findings

The Kaufman projection theorem is sharp for almost Ahlfors regular sets.

Contrasts with recent improvements for strictly Ahlfors regular sets.

Clarifies the limits of projection theorems in different regularity classes.

Abstract

We show that the "sharp Kaufman projection theorem" from 2023 is sharp in the class of Ahlfors $(1,\delta^{-\epsilon})$-regular sets. This is in contrast with a recent result of the first author, which improves the projection theorem in the class of Ahlfors $(1,C)$-regular sets.