Visible parts and lower bounds on point-ray incidences

Abstract

Let $K \subset \mathbb{R}^{2}$ be a compact set. For $θ\in S^{1}$, let $\mathrm{Vis}_θ(K) \subset K$ be the visible part of $K$ in direction $θ$. We prove that $\operatorname{dim}_{\mathrm{H}} \mathrm{Vis}_θ(K) \leq \tfrac{3}{2}$ for $\mathcal{H}^{1}$ almost every $θ\in S^{1}$. The previous record was $\operatorname{dim}_{\mathrm{H}}\mathrm{Vis}_θ(K) \leq 11/6 \approx 1.833$, due to D. Dąbrowski. Our main tool is a variant of a recent incidence lower bound theorem due to Cohen, Pohoata, and Zakharov where, roughly speaking, lines have been replaced by rays, and $δ^{\varepsilon}$-separated incidences are replaced by $1$-separated incidences.