Intersections of fractal sets with rectifiable curves

Abstract

Let $K \subset\mathbb{R}^d$, $d\ge2$, be a connected compact set and $ϕ(t)=tL(t)$ be a gauge function with $L$ non-increasing. We study intersections of $K$ with rectifiable curves and chord-arc curves. We show that if $\int_0^1\frac{dt}{ϕ(t)}<\infty$, then there exists a rectifiable curve $γ\subset\mathbb{R}^d$ such that $\mathcal{H}^ϕ(K \capγ)>0$. Moreover, for every $0<s<1$, there exists a chord-arc curve $γ$ such that $\textrm{H$.$dim}(K \capγ)\ge s$. Conversely, we show that if $K$ is uniformly porous on lines and $\int_0^1\frac{dt}{ϕ(t)}=\infty$, then $\mathcal{H}^ϕ(K \capγ)=0$ for every rectifiable curve $γ$ and $\textrm{H$.$dim}(K \capγ)<1$ for every chord-arc curve $γ$.