REVIEW 2 major objections 3 minor 1 cited by
Improved bounds for radial projections in the plane
T0 review · 2 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash
Pith's one-line read For any two Borel sets in R², some radial projection has dimension at least (dim X + dim Y)/2 unless a simpler bound already applies.
desk verdict Abstract-only look at a clean, sharp radial projection bound; the strong endpoint regime is exactly where the proof needs the most scrutiny. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the radial projection map π_x: Y → S¹, which sends each y to the direction of the ray from x through y. The theorem is a statement about the entire family of these maps indexed by centers x∈X. The minimum formula reflects the trivial upper bound dim_H(π_x Y) ≤ min{dim_H Y, 1}, so the proof's task is to show that, except in the hard triangle, this ceiling is reached, and in the hard triangle that the average is a floor. The abstract does not display the underlying incidence estimate, but the sum-of-dimensions in the bound is characteristic of a discretized multi-scale counting argument that controls how many pairs (x,y) produce directions lying in a narrow angular cone.
What would settle it
Construct two Borel sets X,Y ⊂ R², with X not contained in a line and dim_H(X) = s, dim_H(Y) = t satisfying t > s and t + s < 2, for which sup_{x∈X} dim_H(π_x Y) < (t + s)/2. Such a pair would directly contradict the theorem.
Extended reading notes
Core claim
The paper's theorem, stated in the abstract, is that for Borel sets X,Y ⊂ R² with X not contained in any line and dim_H(X) > 0, sup_{x∈X} dim_H(π_x Y) ≥ min{(dim_H(Y) + dim_H(X))/2, dim_H(Y), 1}. In words: looking at all radial projections of Y from centers in X, at least one of them has Hausdorff dimension as large as the minimum of the two individual dimensions, the average of the two, and 1—whichever is smallest. Since dim_H(π_x Y) ≤ min{dim_H(Y), 1} is the trivial upper bound, the theorem says this ceiling is actually attained whenever dim_H(Y) ≤ dim_H(X), and also whenever the two dimensions sum to at least 2. The only region where the guaranteed value falls below the upper bound is the
Load-bearing premise
The proof's load-bearing premise is a multi-scale incidence estimate controlling how many pairs (x,y) produce directions inside a narrow angular cone, and the abstract does not state or locate that estimate; if it fails at some scale, the average bound (dim X + dim Y)/2 does not follow.
Editorial extensions
If this is right
- Whenever dim_H(Y) ≤ dim_H(X), the theorem forces sup_x dim_H(π_x Y) = dim_H(Y), fully closing the problem in that regime and matching the trivial upper bound.
- Whenever dim_H(X) + dim_H(Y) ≥ 2, some radial projection from X has full dimension 1, meaning the directions from some center form a dimension-1 subset of the circle.
- The only dimension pairs not already settled at the upper bound are those with dim_H(X) < dim_H(Y) and dim_H(X) + dim_H(Y) < 2; there the guarantee is exactly the arithmetic mean of the two dimensions.
- The bound is monotone in both dimensions, so passing to larger Borel supersets cannot reduce the guaranteed projection dimension.
Reading between the lines
- The formula's sharpness in the dim_H(Y) ≤ dim_H(X) regime hints that the true supremum might actually be min{dim_H(Y), 1} for all X with positive dimension, and the average in the hard triangle could be an artifact of the current incidence estimate.
- A natural testable extension is whether the same phase-transition formula carries over to radial projections in R^n onto S^{n−1}, with the 1 replaced by the sphere dimension n−1.
- The assumption dim_H(X) > 0 appears necessary: a zero-dimensional set of centers likely cannot force any projection to be large, so the threshold at dim_H(X) = 0 is a sharp corner.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper, as available to the referee, consists solely of an abstract. It claims a universal, parameter-free theorem: for Borel sets X,Y in R^2 with X not contained in a line and dim_H(X)>0, sup_{x in X} dim_H(pi_x Y) >= min{(dim_H(Y)+dim_H(X))/2, dim_H(Y), 1}. The abstract gives no proof, no derivation, and no statement of any supporting lemma or incidence estimate.
Significance. If true, the claimed bound is significant and internally coherent. In the regime dim_H(Y) <= dim_H(X), it yields sup_x dim_H(pi_x Y) = dim_H(Y), saturating the trivial upper bound and asserting that every Borel X of sufficiently large dimension not contained in a line contains a point from which the radial projection of Y preserves the full dimension of Y. In the regime dim_H(X)+dim_H(Y) >= 2, it gives the maximum possible value 1. In the intermediate regime it gives (dim_H(X)+dim_H(Y))/2, which the authors state improves the best known lower bound. The statement is falsifiable and has no free parameters. However, the strength of the middle regime and the sharp endpoint depend on a multi-scale incidence estimate that is completely absent from the submitted text. A mathematical result of this type cannot be assessed without the proof.
major comments (2)
- [Abstract (theorem statement)] The central result is stated without any supporting argument. The manuscript contains no lemmas, no derivation, and no indication of where the proof is located. In this area, a bound of the form (dim_H(Y)+dim_H(X))/2 typically follows from a multi-scale incidence estimate controlling the number of pairs (x,y) in X x Y whose connecting directions lie in a narrow angular window. Neither the estimate nor its proof is stated. This omission is load-bearing: without the incidence estimate, the theorem cannot be verified.
- [Abstract (endpoint case dim_H(Y) <= dim_H(X))] In the endpoint case, the theorem asserts sup_x dim_H(pi_x Y) = dim_H(Y), meaning full dimension preservation. This is a strong structural statement about every Borel set X with dim_H(X) >= dim_H(Y) that is not contained in a line. The abstract gives no indication of how the reduction from arbitrary Borel sets to compact, doubling, or Frostman-regular sets is performed, nor what quantitative control is needed at every scale. If the approximation step requires additional hypotheses, the universal statement would fail. This needs to be spelled out in the proof.
minor comments (3)
- [Abstract (notation)] The quantities dim_H and pi_x are not explicitly defined. They are standard in the field, but definitions would make the abstract self-contained.
- [Abstract (comparison with prior work)] The phrase 'improve the best known lower bound' is not accompanied by the previous record or a comparison. The authors should state the prior bound and the regime in which the new bound is strictly better.
- [General] The submission contains no references. For a complete paper, references to prior work on radial projections are expected.
Circularity Check
No circularity detectable from the abstract; the claimed theorem is a parameter-free universal statement with no fitted inputs or self-citation chain.
full rationale
The only available text is the abstract, which states a universal lower bound for all Borel sets X,Y in R^2 under explicit hypotheses (X not contained in any line, dim_H(X)>0). The claimed inequality is not derived from any fitted parameter, nor is any 'prediction' computed from data, and no self-citation appears in the abstract. There is no equation in the abstract that reduces to its own inputs by definition. The concern raised in the reader's take — that the proof likely depends on an unstated multi-scale incidence estimate and that the full proof is unavailable — is a correctness/verifiability issue, not a circularity issue: an unstated lemma cannot be shown, from the abstract alone, to be equivalent to the theorem by construction. Under the hard rule that circularity must be exhibited by quoting the paper and showing a specific reduction, no circular step can be identified. Therefore the honest finding is no significant circularity, score 0.
Assumptions & free parameters
assumptions (4)
- standard math Hausdorff dimension and Borel measurability in R^2 are used with their standard definitions
- domain assumption The hypotheses that X is not contained in any line and dim_H(X) > 0 are essential to rule out degenerate center sets
- standard math Radial projection cannot increase Hausdorff dimension: dim(pi_x Y) <= dim_H(Y) for every x
- domain assumption A reduction from general Borel sets to discretized Frostman measures is valid at all scales (inferred)
Cite this review
Pith. "Pith review of Improved bounds for radial projections in the plane." pith.science (2026). https://pith.science/paper/6ALZ7IEO
@misc{pith2026250818228,
author = {Pith},
title = {Pith review of: Improved bounds for radial projections in the plane},
year = {2026},
howpublished = {\url{https://pith.science/paper/6ALZ7IEO}},
note = {Machine review of arXiv:2508.18228}
}
abstract
We improve the best known lower bound for the dimension of radial projections of sets in the plane. We show that if $X,Y$ are Borel sets in $\R^2$, $X$ is not contained in any line and $\dim_H(X)>0$, then $$\sup\limits_{x\in X} \dim_H(\pi_x Y) \geq \min\left\{(\dim_H(Y) + \dim_H(X))/2, \dim_H(Y), 1\right\},$$ where $\pi_x Y$ is the radial projection of the set $Y$ from the point $x$.
Forward citations
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Reviewed August 5, 2026 · model on record in the stance chip above.
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