On the dimension and smoothness of radial projections

By Tuomas Orponen. Analysis & PDE, 2017

55 citations recorded by the source corpus.

This paper contains two results on the dimension and smoothness of radial projections of sets and measures in Euclidean spaces. To introduce the first one, assume that $E,K \subset \mathbb{R}^{2}$ are non-empty Borel sets with $\dim_{\mathrm{H}} K > 0$. Does the radial projection of $K$ to some point in $E$ have positive dimension? Not necessarily: $E$ can be zero-dimensional, or $E$ and $K$ can lie on a common line. I prove that these are the only obstructions: if $\dim_{\mathrm{H}} E > 0$, and $E$ does not lie on a line, then there exists a point in $x \in E$ such that the radial projection $\pi_{x}(K)$ has Hausdorff dimension at least $(\dim_{\mathrm{H}} K)/2$. Applying the result with $E = K$ gives the following corollary: if $K \subset \mathbb{R}^{2}$ is Borel set, which does not lie on a line, then the set of directions spanned by $K$ has Hausdorff dimension at least $(\dim_{\mathrm{H}} K)/2$. For the second result, let $d \geq 2$ and $d - 1 1$. The dimension bound on the exceptional set is sharp.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5bdcd0828fde84d82f5c34464b10589694d7f92e

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In Applied Mathematics

In Computer Vision and Pattern Recognition

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Citations in the corpus

New Bounds on the Dimensions of Planar Distance Sets (2018, background): “In fact, we need a far more quantitative “pinned” version of this fact, which is due to Orponen [21], improving upon a related result by Mattila and Orponen [19] (see Proposition 3.”
Weighted restriction estimates and application to Falconer distance set problem (2018, background): “We also point the interested reader to [12, 19, 21, 24] and the references therein for more results along this line of research.”
Incidence Estimates for Well Spaced Tubes (2019, methodology): “This Fourier analysis argument is based on arguments in the literature on projection theory, especially the recent paper by Orponen [17].”
On Falconer’s distance set problem in the plane (2018, methodology): “general a point xcould lie in many bad tubes, and we need to know that the total 1 measure of all these bad tubes is small (for most xPE2). This follows from Orponen’s radial projection theorem from [31]. This theorem plays an important role in Keleti and Shmerkin’s work on the Falconer problem [25]”
A sharp square function estimate for the cone in ℝ3 (2019, methodology): “This approach to proving Kakeya-type estimates is based on some work of Orponen in projection theory [21] and is related to Vinh’s work [25] about incidence geometry over finite fields.”