On the Hausdorff dimension of radial slices

Let \ (t (1, 2) \), and let \ (B R^2\) be a Borel set with \ (₇ B > t\). I show that \ (H^1 (\e S^{1 ₇ (B ₗ, ₄) t - 1\}) > 0\) for all \ (x R^2 E\), where \ (₇ E 2 - t\). This is the sharp bound for \ (₇ E\). The main technical tool is an incidence inequality of the form \ (I_ (, ) ₓ Iₓ () ₈_₃ - ₓ () \), \ (t (1, 2) \), where \ (\) is a Borel measure on \ (R^2\), and \ (\) is a Borel measure on the set of lines in \ (R^2\), and \ (I_ (, ) \) measures the \ (\) -incidences between \ (\) and the lines parametrised by \ (\). This inequality can be viewed as a \ (^-\) -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 \ (^-\) -factor in Fu and Ren's work. Instead, the inequality is deduced from the classical smoothing properties of the \ (X\) -ray transform.