Improved bounds for radial projections in the plane

Abstract

We improve the best known lower bound for the dimension of radial projections of sets in the plane. We show that if $X,Y$ are Borel sets in ${\mathbb{R}}^2$, $X$ is not contained in any line and ${\dim }_H(X)>0$, then $$\underset{x\in X}{\sup }{\dim }_H(\ddot{I}{€}_{x}Y)\ge \min \left\{({\dim }_H(Y)+{\dim }_H(X))/2,{\dim }_H(Y),1\right\},$$ where $\ddot{I}{€}_{x}Y$ is the radial projection of the set $Y$ from the point $x$.