REVIEW 2 major objections 5 minor 1 cited by
A one-dimensional planar Besicovitch-type set
T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read A construction of a nontrivial 1-rectifiable set Γ with a 1-dimensional Γ-Besicovitch set.
desk verdict The construction fails at the first step: the sequence conditions in Section 2 are inconsistent, so Γ and B are never defined. 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 a nested family of rectangles $Q_{n,j}$ with $x$-projection length $\delta_n$ and $y$-projection close to $\Delta_n$, together with a rule that replaces each rectangle by smaller rectangles positioned at points $p_{n,j,k}$ obtained by rotating the corner $p_{n,j}$ about a centre $\alpha_{n+1}$ on a circular arc through multiples of the angle $\theta_{n+1} = c\Delta_{n+1}\delta_n$. The identity $p_{n,j,k+l} = e^{-il\theta_{n+1}}p_{n,j,k} + p_{n,1,l+1}$ makes the point configuration invariant under a rotation-and-translation step, and that self-overlap is what lets the $\Gamma$-Besicovitch set $B = \bigcap_{n}\bigcup_{l \le \theta_n^{-1}} T_{n,l\theta_n}$ be shown to have Hausdorff dimension 1.
What would settle it
Check the parameter conditions at $n=1$: from $\Delta_2 \le c\delta_2$ and $\delta_2 \le c\delta_1\Delta_2 = c\Delta_2$, multiplying gives $\Delta_2\delta_2 \le c^2\Delta_2\delta_2$, hence $c^2 \ge 1$, contradicting $c<1$. Exhibiting a valid pair of sequences satisfying all four conditions with some $c<1$ would refute this check; otherwise the construction as stated cannot be instantiated.
Extended reading notes
Core claim
This paper's central claim is the existence of a nontrivial 1-rectifiable planar set $\Gamma$ with positive 1-dimensional Hausdorff measure that is the graph of a monotone function over a measure-zero Cantor set, and for which there is a $\Gamma$-Besicovitch set of Hausdorff dimension 1. The Cantor domain can be prescribed to have any Hausdorff dimension $s \in [0,1]$. The proof constructs $\Gamma$ as the countable intersection of nested sets $S_n$, each a finite union of axis-parallel rectangles with $x$-length $\delta_n$ and $y$-length close to $\Delta_n$, and the dimension-1 set $B$ is defined as a nested union of rotated, translated thickened rectangles. The argument revolves around the identity $p_{n,j,k+l} = e^{-il\theta_{n+1}}p_{n,j,k} + p_{n,1,l+1}$, which creates systematic overlaps between the rotated copies and controls the measure of thin neighbourhoods of $B$.
Load-bearing premise
The construction assumes one can fix a constant $c<1$ and positive decreasing sequences $\delta_n,\Delta_n \to 0$ satisfying $\delta_1=\Delta_1=1$, $\Delta_{n+1} \le c\delta_{n+1}^n$, $\delta_{n+1} \le c\Delta_{n+1}\delta_n$, and an integrality condition; every later estimate relies on these inequalities, so if no such sequences exist the construction cannot begin.
Editorial extensions
If this is right
- A valid construction would answer the open question affirmatively: a nontrivial 1-rectifiable set can have a $\Gamma$-Besicovitch set of dimension 1.
- The example would lie outside the known circular-arc family, since $\Gamma$ has positive length and parallel tangents almost everywhere and so is not covered by a low-dimensional union of concentric circles.
- The parameter $s \in [0,1]$ gives a continuum of Cantor-graph domains, so the dimension of the domain can be tuned without forcing the $\Gamma$-Besicovitch dimension above 1.
- The result would separate this monotone Cantor-graph case from the $C^3$ non-circular arc case, where the $\Gamma$-Besicovitch set is known to have dimension 2.
Reading between the lines
- Not made in the paper: the parameter conditions (2) and (3) should be checked for consistency. At $n=1$ they imply $\Delta_2 \le c\delta_2$ and $\delta_2 \le c\Delta_2$, so $c^2 \ge 1$, contradicting $c<1$; if no admissible sequences exist, the construction as written has no starting point.
- If one could repair the parameter choice, the same rotation-translation identity suggests a recipe for other sparse graphs whose tangents are vertical on a null set, possibly yielding $\Gamma$-Besicovitch sets of intermediate dimension.
- The dimension-tuning equation $\delta_n = c\Delta_n^{1/s_n}$ indicates a one-parameter family of examples; testing whether the mechanism survives with slower decay of $\delta_n$ would clarify where the dimension-1 phenomenon ends.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to construct a nontrivial 1-rectifiable planar set Γ with positive Hausdorff dimension for which there exists a 1-dimensional Γ-Besicovitch set, answering a question of Csörnyei. The construction defines Γ as an intersection of nested unions of rectangles governed by two positive decreasing sequences {δ_n} and {∆_n} satisfying conditions (1)–(4) in Section 2. Subsequent sections estimate the number and measure of rectangles, build a candidate Besicovitch set B, and prove that the domain of Γ can have any prescribed Hausdorff dimension s ∈ [0,1]. The paper is organized around a detailed inductive construction with several lemmas (Lemmas 2–7) and a dimension computation in Section 4.
Significance. If the construction were valid, it would provide a significant affirmative answer to a well-known open question, moving beyond circle-covered examples and giving a 1-rectifiable set with parallel tangents almost everywhere that admits a 1-dimensional Γ-Besicovitch set. The paper merits credit for a concrete and ambitious inductive strategy, for clearly situating the problem relative to previous work by Chang–Csörnyei and others, and for including explicit estimates in Lemmas 3–6. However, the central sequence conditions are mutually inconsistent, so the sets Γ and B are, as defined, non-existent; this prevents the main theorem from being established.
major comments (2)
- [Section 2, equations (2)–(3)] The defining conditions for the sequences {δ_n} and {∆_n} are inconsistent for any c < 1. Setting n = 1 in (2) gives ∆_2 ≤ cδ_2, while (3) with n = 1 gives δ_2 ≤ c∆_2δ_1 = c∆_2. Combining these yields δ_2 ≤ c^2δ_2, and since δ_2 > 0, this forces c ≥ 1, contradicting the stated c < 1. Therefore no positive decreasing sequences satisfying (1)–(3) exist, so the rectangles Q_{n,j}, the nested sets S_n, and Γ are never defined. This invalidates Theorem 1, and all subsequent estimates that rely on these sequences (Lemmas 2–4 and the dimension analysis in Section 4) rest on a false premise.
- [Section 3, Lemma 5] The proof of the dimension bound contains an unsupported inference. The text states 'From (2), we see that Δ_{n+1}^{1/(n+1)} ≤ c^{1/(n+1)} δ_n', but (2) is Δ_{n+1} ≤ cδ_{n+1}^n, which gives Δ_{n+1}^{1/(n+1)} ≤ c^{1/(n+1)} δ_{n+1}^{n/(n+1)}, not a bound by δ_n. This inequality is used to obtain the containment B(cΔ_{n+1}^{1+1/(n+1)}) ⊂ B(θ_{n+1}) and hence the estimate dim_H B = 1. The step can potentially be repaired using monotonicity and δ_n ≤ 1, but it is not justified as written, and the proof of Lemma 5 currently has a gap at exactly the point where the dimension conclusion is drawn.
minor comments (5)
- [Throughout] There are several typographical issues: 'For each rectangle inSn' should read 'in S_n', 'monotonous' should be 'monotone', and '1-dimenionsal set' in Lemma 5 should be '1-dimensional set'.
- [Section 3, equation (32)] The symbol Q_{n,j} is reused: earlier it denotes the rectangle with bottom-left corner p_{n,j}, while in (32) it denotes a larger rectangle centered at p_{n,j}. This notation clash makes the containment statement 'Q_{n,j} centered at p_{n,j} that contain Q_{n,j}' difficult to parse.
- [Section 3, equation (37)] The union in (37) is written as ⋃_{K<θ_nθ_{n+1}^{-1}} T'_{n+1,lθ_{n+1}}, but the subscript of T' should be Kθ_{n+1} rather than lθ_{n+1}; the label is inconsistent with the preceding definition.
- [Section 4, verification of (3) from (48)] The chain showing that (48) implies (3) is terse; adding the intermediate inequality δ_{n+1}^{n(1/s_{n+1}-1)} ≤ δ_n^{n/(n+1)} ≤ δ_n would make the argument easier to follow.
- [Section 4, proof of dim(P_x(Γ)) ≤ s] In the case s = 0, the estimate δ_p^{s'-1/p} < 1 is asserted for p such that 1/p < s'; since δ_p ≤ 1, the bound is correct, but the sentence should state explicitly that s' > 1/p is used to make the exponent negative.
Circularity Check
No circular reasoning: the construction is self-contained. The fatal flaw is an unsatisfiable premise — conditions (2)-(3) at n=1 force c ≥ 1, contradicting c < 1 — a soundness failure, not circularity.
full rationale
The derivation chain is self-contained and contains no circular step. Section 2 constructs Γ from scratch: the rectangles Q_{n,j,k} are defined by explicit formulas (9)–(14) from the assumed sequences δ_n, Δ_n; Lemma 2 derives the separation estimates from the circle geometry and the definition θ_{n+1} = cΔ_{n+1}δ_n; Lemma 3 derives the counting bounds and the positive-measure conclusion (23) purely from these lemmas; Section 3 proves dim_H B = 1 in Lemma 5 by a self-contained measure estimate on the sets B(θ_{n+1}) built from (34); and Section 4 fixes the free parameters via (48) and proves dim(Px(Γ)) = s using Lemma 3. No fitted quantity is renamed as a prediction, no uniqueness theorem is imported, and the only self-citation ([4]) is contextual motivation, not a load-bearing premise, so none of the seven circularity patterns is present. The paper's genuine defect is flagged here as missing support: Section 2 asserts the existence of positive decreasing sequences satisfying (1) δ_1 = Δ_1 = 1, (2) Δ_{n+1} ≤ cδ_{n+1}^n, and (3) δ_{n+1} ≤ cΔ_{n+1}δ_n with a fixed c < 1. Substituting n = 1 gives Δ_2 ≤ cδ_2 and δ_2 ≤ cΔ_2, hence δ_2 ≤ c^2δ_2, which with δ_2 > 0 forces c ≥ 1, contradicting c < 1. (Section 4's choice (48) δ_n = cΔ_n^{1/s_n} likewise forces δ_1 = c ≠ 1, contradicting (1).) No such sequences exist, so Γ and B are undefined and Theorem 1 is not established. This is an empty-premise failure — the results reduce to nothing rather than to their own inputs — so it is soundness, not circularity, and leaves the circularity score at 0.
Assumptions & free parameters
free parameters (1)
- c =
any value in (0, 1/10)
assumptions (4)
- ad hoc to paper There exist positive decreasing sequences {δ_n}, {Δ_n} converging to 0 and satisfying (1) δ_1 = Δ_1 = 1, (2) Δ_{n+1} ≤ cδ_{n+1}^n, (3) δ_{n+1} ≤ cΔ_{n+1}δ_n, and (4) integrality conditions, for a fixed c < 1.
- standard math Every 1-rectifiable set in the plane has a tangent field defined almost everywhere.
- standard math A circular arc with center below the x-axis is the graph of a concave monotone function.
- standard math If |B(r)| ≲ r^{1-ε} for arbitrarily small r, then dim_H B ≤ 1+ε.
Cite this review
Pith. "Pith review of A one-dimensional planar Besicovitch-type set." pith.science (2026). https://pith.science/paper/YZKW7ODB
@misc{pith2026241211190,
author = {Pith},
title = {Pith review of: A one-dimensional planar Besicovitch-type set},
year = {2026},
howpublished = {\url{https://pith.science/paper/YZKW7ODB}},
note = {Machine review of arXiv:2412.11190}
}
abstract
A $\Gamma$-Besicovitch set is a set which contains a rotated copy of $\Gamma$ in every direction. Our main result is the construction of a non-trivial $1$-rectifiable set $\Gamma$ in the plane, for which there exists a 1-dimensional $\Gamma$-Besicovitch set.
Figures
Forward citations
Cited by 1 Pith paper
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A short note on the dimension of the CC-geodesic Kakeya sets in the first Heisenberg group
CC-geodesic Kakeya sets in the first Heisenberg group have sharp Heisenberg Hausdorff dimension 3 (even when compact), not the predicted 4.
Reference graph
Works this paper leans on
-
[1]
The Kakeya needle problem and the existence of Besicovitch and Nikodym sets for rectifiable sets,
A. Chang and M. Cs¨ ornyei, “The Kakeya needle problem and the existence of Besicovitch and Nikodym sets for rectifiable sets,” Proc. Lond. Math. Soc., vol. 118, no. 5, pp. 1084–1114, 2019
work page 2019
-
[2]
A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in R3,
M. Pramanik, T. Yang, and J. Zahl, “A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in R3,” arXiv:2207.02259
-
[3]
On the Wolff circular maximal function,
J. Zahl, “On the Wolff circular maximal function,” Illinois J. Math, vol. 56, no. 4, pp. 1281–1295, 2014
work page 2014
-
[4]
Hausdorff dimension of Besicovitch sets of Cantor graphs,
I. Altaf, M. Cs¨ ornyei, and K. H´ era, “Hausdorff dimension of Besicovitch sets of Cantor graphs,”Mathe- matika, vol. 70, p. e12241, 2024
work page 2024
-
[5]
Closed sets with the Kakeya property,
M. Cs¨ ornyei, K. H´ era, and M. Laczkovich, “Closed sets with the Kakeya property,”Mathematika, vol. 63, pp. 184–195, 2017
work page 2017
-
[6]
K. J. Falconer, “The geometry of fractal sets,” Cambridge University Press, 1985
work page 1985
-
[7]
Hausdorff dimension of unions of affine subspaces and of Furstenberg- type sets,
K. H´ era, T. Keleti, and A. M´ ath´ e, “Hausdorff dimension of unions of affine subspaces and of Furstenberg- type sets,” J. Fractal Geom., vol. 6, pp. 263–284, 2019
work page 2019
-
[8]
Mattila, Geometry of sets and measures in Euclidean spaces
P. Mattila, Geometry of sets and measures in Euclidean spaces. Cambridge University Press, 1995
work page 1995
Show all 9 references
-
[9]
The Kakeya problem for circular arcs,
K. H´ era and M. Laczkovich, “The Kakeya problem for circular arcs,” Acta Mathematica Hungarica, vol. 150, pp. 479–511, 2016. Department of Mathematics, The University of Chicago, 5734 S. University A venue, Chicago, IL 60637, USA. email: iqra@uchicago.edu. 16
2016
Reviewed August 11, 2026 · model on record in the stance chip above.
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