On the Hausdorff dimension of radial slices

Abstract

Let $t \in (1,2)$, and let $B \subset \mathbb{R}^{2}$ be a Borel set with $\dim_{\mathrm{H}} B > t$. I show that $$\mathcal{H}^{1}(\{e \in S^{1} : \dim_{\mathrm{H}} (B \cap \ell_{x,e}) \geq t - 1\}) > 0$$ for all $x \in \mathbb{R}^{2} \, \setminus \, E$, where $\dim_{\mathrm{H}} E \leq 2 - t$. This is the sharp bound for $\dim_{\mathrm{H}} E$. The main technical tool is an incidence inequality of the form $$\mathcal{I}_{\delta}(\mu,\nu) \lesssim_{t} \delta \cdot \sqrt{I_{t}(\mu)I_{3 - t}(\nu)}, \qquad t \in (1,2),$$ where $\mu$ is a Borel measure on $\mathbb{R}^{2}$, and $\nu$ is a Borel measure on the set of lines in $\mathbb{R}^{2}$, and $\mathcal{I}_{\delta}(\mu,\nu)$ measures the $\delta$-incidences between $\mu$ and the lines parametrised by $\nu$. This inequality can be viewed as a $\delta^{-\epsilon}$-free version of a recent incidence theorem due to Fu and Ren. The proof in this paper avoids the high-low method, and the induction-on-scales scheme responsible for the $\delta^{-\epsilon}$-factor in Fu and Ren's work. Instead, the inequality is deduced from the classical smoothing properties of the $X$-ray transform.