On the Hausdorff dimension of radial slices
Let \(t (1,2)\), and let \(B R^2\) be a Borel set with \(_H B > t\). I show that \(H^1({e S^1 _H (B _x,e) ≥ t - 1}) > 0\) for all \(x R^2 E\), where \(_H E ≤ 2 - t\). This is the sharp bound for \(_H E\). The main technical tool is an incidence inequality of the form \(I_δ(μ, u) _t δ · I_t(μ)I_3 - t( u)\), \(t (1,2)\), where \(μ\) is a Borel measure on \(R^2\), and \( u\) is a Borel measure on the set of lines in \(R^2\), and \(I_δ(μ, u)\) measures the \(δ\)-incidences between \(μ\) and the lines parametrised by \( u\). This inequality can be viewed as a \(δ^-ε\)-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 \(δ^-ε\)-factor in Fu and Ren’s work. Instead, the inequality is deduced from the classical smoothing properties of the \(X\)-ray transform. , number=1, journal=Annales Fennici Mathematici, author=Orponen, Tuomas, year=2024, month=Mar., pages=183–209