On the projections of Ahlfors regular sets in the plane

Abstract: This paper contains the following $\delta$-discretised projection theorem for Ahlfors regular sets in the plane. For all $C,\epsilon &gt; 0$ and $s \in [0,1]$, there exists $\kappa &gt; 0$ such that the following holds for all $\delta &gt; 0$ small enough. Let $\nu$ be a Borel probability measure on $S<sup>{1}$ satisfying $\nu(B(x,r)) \leq Cr<sup>{\epsilon}$ for all $x \in S<sup>{1}$ and $r &gt; 0$. Let $K \subset B(1) \subset \mathbb{R}<sup>{2}$ be Ahlfors $s$-regular with constant at most $C$. Then, there exists a vector $\theta \in \mathrm{spt\,} \nu$ such that $$|\pi_{\theta}(F)|<em>{\delta} \geq \delta^{\epsilon - s}$$ for all $F \subset K$ with $$|F|</em>{\delta} \geq \delta^{\kappa - s}$$. Here $\pi_{\theta}(z) = \theta \cdot z$ for $z \in \mathbb{R}<sup>{2}$.