On Hausdorff dimension of radial projections (Q2039469)

Given a Borel set \(E\subset \mathbb{R}^d\) with Hausdorff dimension \(\mathrm{dim}_{\mathcal{H}} E\in (d-2, d-1\,]\), the author of the present paper studies the Hausdorff dimension of the radial projections of \(E\). The radial projection from a point \(x\in \mathbb{R}^d\) is the map \[ \pi^x: \mathbb{R}^d\backslash \{x\} \to \mathrm{S}^{d-1},\ y \mapsto \displaystyle\frac{y-x}{|y-x|}. \] The author studies the set of points \(x\) such that \[ \mathrm{dim}_{\mathcal{H}} \pi^x(E) < \mathrm{dim}_{\mathcal{H}}E \] and shows that the Hausdorff dimension of this set is at most \(2(d-1)-\mathrm{dim}_{\mathcal{H}}E\). This improves the previously known bounds in the range \(\mathrm{dim}_{\mathcal{H}}E \in (d-3/2,d-1\,]\), the latter bounds can be obtained by using the machinery of of Peres and Schlag ( Theorem 7.3 in [\textit{Y. Peres} and \textit{W. Schlag}, Duke Math. J. 102, No. 2, 193--251 (2000; Zbl 0961.42007)]) as radial projections form a transverse family. The main step of the proof (section 3 of the paper) involves an interesting symmetry argument. It is shown that given a set \(F\), the existence of a point \(x\in F\) such that \(\mathrm{dim}_{\mathcal{H}} \pi^x(E)\ge \tau \in(0, \mathrm{dim}_{\mathcal{H}} E)\) follows from an estimate on the radial projections of \(F\) from points in \(E\). In section 4, the author then shows that this estimate holds for all \(\tau\) whenever \(\mathrm{dim}_{\mathcal{H}} F > 2(d-1) - \mathrm{dim}_{\mathcal{H}}E\, (\ge d-1)\) by using results from the paper [\textit{T. Orponen}, Anal. PDE 12, No. 5, 1273--1294 (2019; Zbl 1405.28011)], in which T.~Orponen studies the radial projections of sets with dimension larger than \(d-1\).