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On the dimension and smoothness of radial projections
Classical Analysis and ODEs · Submitted 30 Oct 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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 $(\dim_{\mathrm{H}} K)/2$.
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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 $d - 1 < s < d$. Let $\mu$ be a compactly supported Radon measure in $\mathbb{R}^{d}$ with finite $s$-energy. I prove that the radial projections of $\mu$ are absolutely continuous with respect to $\mathcal{H}^{d - 1}$ for every centre in $\mathbb{R}^{d} \setminus \operatorname{spt} \mu$, outside an exceptional set of dimension at most $2(d - 1) - s$. In fact, for $x$ outside an exceptional set as above, the proof shows that $\pi_{x\sharp}\mu \in L^{p}(S^{d - 1})$ for some $p > 1$. The dimension bound on the exceptional set is sharp.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This paper contains two results on the dimension and smoothness of radial projections of sets and measures in Euclidean spaces. | contribution 0.94 | 0.56 | 0.07 | asserted 0.96 |
| s1 | To introduce the first one, assume that $E,K \subset \mathbb{R}^{2}$ are non-empty Borel sets with $\dim_{\mathrm{H}} K > 0$. | background 0.93 | 0.10 | 0.02 | asserted 0.55 |
| s2 | Does the radial projection of $K$ to some point in $E$ have positive dimension? | background 0.81 | 0.04 | 0.02 | unclear 0.71 |
| s3 | Not necessarily: $E$ can be zero-dimensional, or $E$ and $K$ can lie on a common line. | background 0.73 | 0.13 | 0.03 | asserted 0.69 |
| s4 | 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 $(\dim_{\mathrm{H}} K)/2$. | result 0.97 | 0.96 | 0.52 | asserted 1.00 |
| s5 | 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 $(\dim_{\mathrm{H}} K)/2$. | result 0.88 | 0.89 | 0.13 | asserted 1.00 |
| s6 | For the second result, let $d \geq 2$ and $d - 1 < s < d$. | background 0.80 | 0.12 | 0.02 | unclear 0.75 |
| s7 | Let $\mu$ be a compactly supported Radon measure in $\mathbb{R}^{d}$ with finite $s$-energy. | background 0.81 | 0.12 | 0.02 | asserted 0.71 |
| s8 | I prove that the radial projections of $\mu$ are absolutely continuous with respect to $\mathcal{H}^{d - 1}$ for every centre in $\mathbb{R}^{d} \setminus \operatorname{spt} \mu$, outside an exceptional set of dimension at most $2(d - 1) - s$. | result 0.99 | 0.96 | 0.32 | asserted 1.00 |
| s9 | In fact, for $x$ outside an exceptional set as above, the proof shows that $\pi_{x\sharp}\mu \in L^{p}(S^{d - 1})$ for some $p > 1$. | result 0.93 | 0.93 | 0.11 | asserted 0.99 |
| s10 | The dimension bound on the exceptional set is sharp. | result 0.65 | 0.70 | 0.07 | asserted 1.00 |
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The paper
Original abstract
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 $d - 1 < s < d$. Let $\mu$ be a compactly supported Radon measure in $\mathbb{R}^{d}$ with finite $s$-energy. I prove that the radial projections of $\mu$ are absolutely continuous with respect to $\mathcal{H}^{d - 1}$ for every centre in $\mathbb{R}^{d} \setminus \operatorname{spt} \mu$, outside an exceptional set of dimension at most $2(d - 1) - s$. In fact, for $x$ outside an exceptional set as above, the proof shows that $\pi_{x\sharp}\mu \in L^{p}(S^{d - 1})$ for some $p > 1$. The dimension bound on the exceptional set is sharp.
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How the literature uses it
6 arXiv papers cite this in 15 sentences
Papers whose main result relates to this one's
States the same result 1 paper
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I prove that the radial projections $\pi_{x\sharp}\mu$ of $\mu$ are absolutely continuous with respect to $\mathcal{H}^{d - 1}$ for every centre $x \in \mathbb{R}^{d} \setminus \operatorname{spt} \mu$, outside an exceptional set of dimension at most $2(d - 1) - s$. This is sharp.
Papers citing it · 10 sentences without a confident label
Uses its method or tool 2 papers
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To control this type of behavior, we use an estimate of Orponen from [ref] which also played a key role in Keleti and Shmerkin's work on the Falconer problem [ref] .
On Falconer's distance set problem in the plane · The main obstacle · 0.99
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This Fourier analysis argument is based on arguments in the literature on projection theory, especially the recent paper by Orponen [ref] .
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Compares with it 1 paper
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In [ref] , Orponen also proved that the direction set of a planar set of Hausdorff dimension $1$ which is not contained in a line has Hausdorff dimension $≥1/2$, but this is very far from positive measure, let alone from anything resembling Proposition [ref].
New bounds on the dimensions of planar distance sets · Sets of directions, and the case of dimension $1$ · 0.93
Cites it as background 1 paper
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Also, similar problems have been studied in various other settings such as Heisenberg groups [ref] [ref] [ref] and normed spaces [ref] [ref] , as well as for radial projections in [ref] , different notions of measure and dimension [ref] [ref] [ref] , and restricted families of projections [ref] [ref] [ref] (and references therein).
Projection theorems in hyperbolic space · Introduction · 1.00
What it cites · 2 sentences without a confident label
Compares with 1 paper
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Theorem [ref] does not immediately follow from the proof of [ref] in [ref] and [ref] , as the argument in those papers was somewhat indirect.
A sharp exceptional set estimate for visibility · The second main result · 0.93
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