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
\(\mathcal{H}^{1}(\{e \in S^{1} \colon \dim_{\mathrm{H}} (B \cap \ell_{x,e}) \geq t - 1\}) > 0\)

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
\(\mathcal{I}_{\delta}(\mu,\nu) \lesssim_{t} \delta \cdot \sqrt{I_{t}(\mu)I_{3 - t}(\nu)}\), \(t \in (1,2)\),

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.