On the Hausdorff dimension of radial slices

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 $${\mathcal{H}}^1(\{e\in S^1:{\dim }_{\mathrm{H}}(B\cap {\mathrm{\ell }}_{x,e})\ge t-1\})>0$$ for all $x\in {\mathbb{R}}^2\setminus E$, where ${\dim }_{\mathrm{H}}E\le 2-t$. This is the sharp bound for ${\dim }_{\mathrm{H}}E$. The main technical tool is an incidence inequality of the form $${\mathcal{I}}_{\delta }(\mu ,\nu ){\lesssim }_t\delta \cdot \sqrt{I_t(\mu )I_{3-t}(\nu )},t\in (1,2),$$ 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.