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Applications of Nonlinear Projections to Rectifiable 1-sets

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Pith's one-line read A classical projection theorem for rectifiable sets is extended to nonlinear maps: distance, radial, and circle families with a nondegenerate canonical embedding each force at least one positive-length image, with bad indices in a…

desk verdict A clean, reusable framework for nonlinear projections at the rectifiable endpoint, with two sloppy but fixable WLOG reductions. read the letter →

arxiv 2608.10253 v1 pith:Z4VIMEFM submitted 2026-08-10 math.CA

Rosemarie Bongers , Paige Bright , Caleb Marshall , Krystal Taylor This is my paper · ORCID
classification math.CA MSC 28A7528A78
keywords nonlinearprojectionsFedererprojectiontheorem1-rectifiablesetspinneddistanceradialunionsofcirclescanonicalembeddingexceptional
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Federer's projection theorem says that among any $d$ linearly independent orthogonal projections of a 1-rectifiable set in $\mathbb{R}^d$, at least one image has positive length. This paper claims the same conclusion survives when the linear projections are replaced by nonlinear maps—distance to a pin, angle from a vantage point, or the height of a circle intersection—provided the maps can be assembled into a single embedding whose Jacobian determinant is bounded away from zero. The payoff is a unified proof of three results at the critical dimension 1: every 1-rectifiable set contains a pin whose pinned distance set has positive measure, planar radial projections have at most one bad vantage point unless the set is essentially linear, and variable-radius circles centered on a 1-rectifiable set cover positive area. A curious reader should care because these statements are usually attacked with Fourier or dimension-theoretic tools, while here they reduce to a determinant calculation plus a classical projection theorem.

What carries the argument

The load-bearing object is the canonical embedding $H_{\vec{\alpha}}(z)=(\varphi_{\alpha_1}(z),\dots,\varphi_{\alpha_d}(z))$ built from a family of nonlinear scalar maps. The proofs work by computing $\det DH_{\vec{\alpha}}$ and showing it is bounded away from zero and infinity on a conveniently chosen slab or strip: for distance-squared pins the determinant is $2^d\,\pi_d(z)\det[p_1-p_2\,\cdots\,p_1-p_d]$, for radial angle maps it is $y/((x_+^2+y^2)(x_-^2+y^2))$, and for circle intersections it is a sum of three separately controlled terms. Boundedness away from zero makes $H_{\vec{\alpha}}$ locally bilipschitz, hence positivity-preserving for length, and converts the nonlinear problem into the linear projection problem solved by Federer's theorem. The black-box use of that theorem is deliberate: all technical difficulty is relocated into the Jacobian estimate.

What would settle it

A direct disproof would be a 1-rectifiable planar set $E$, not essentially a line, with two distinct vantage points $p\neq q$ for which $H^1(\pi_p(E))=H^1(\pi_q(E))=0$, since Theorem 3.1 permits at most one such point. A natural test case is a union of three non-collinear segments: the radial image from the common endpoint is a finite set, and Theorem 3.1 predicts every other point is good, so checking the radial image from a second segment endpoint would already put the mechanism at risk.

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Extended reading notes

Core claim

The central claim is a transfer principle. Given indices $\alpha_1,\dots,\alpha_d$ and maps $\varphi_{\alpha_j}:\Omega\to\mathbb{R}$, form the canonical embedding $H_{\vec{\alpha}}(z)=(\varphi_{\alpha_1}(z),\dots,\varphi_{\alpha_d}(z))$. If $H_{\vec{\alpha}}$ is bilipschitz—or more generally if $\det DH_{\vec{\alpha}}$ is bounded away from zero and infinity on the relevant domain—then for every 1-rectifiable set $E\subseteq\Omega$ with positive length, $H_{\vec{\alpha}}(E)$ is again a 1-rectifiable set of positive length. Applying the classical projection theorem to $H_{\vec{\alpha}}(E)$ forces at least one coordinate projection $\pi_j(H_{\vec{\alpha}}(E))$ to have positive length, and that coordinate projection is exactly the nonlinear image $\varphi_{\alpha_j}(E)$. The paper realizes this scheme for distance-squared maps $\varphi_p(z)=|z-p|^2$, for planar arctangent angle maps, and for circle-intersection maps $\varphi_\alpha(x,y)=y+\sqrt{r(z)^2-(\alpha-x)^2}$, then converts 'at least one good index' into sharp geometric restrictions on the exceptional set of bad pins or vantage points.

Load-bearing premise

The proofs for pinned distances and radial projections assume, without proof, that a 1-rectifiable set of positive length can be cut down to a bounded piece, still of positive length, on which the encoding map is one-to-one with stretch factors bounded above and below; if this cutting-down step fails, the determinant estimate alone does not imply that the image has positive length.

Editorial extensions

If this is right

  • Every 1-rectifiable set $E\subseteq\mathbb{R}^d$ with $H^1(E)>0$ contains a pin $p$ with $H^1(\Delta_p(E))>0$, and the set of bad pins lies in an affine subspace of dimension at most $d-2$.
  • In the plane, a 1-rectifiable set that is not essentially 1-flat has at most one bad radial vantage point; an essentially 1-flat set has bad vantage points forming exactly a line.
  • For any admissible radius function $r$—bounded above and below, differentiable with bounded gradient, and with monotone partial derivatives—the union of circles centered on a 1-rectifiable set $E\subseteq\mathbb{R}^2$ has positive two-dimensional Lebesgue measure.
  • The framework itself yields a nonlinear d-lines theorem: any $d$ nonlinear maps whose canonical embedding satisfies the Jacobian condition contain at least one good index, and this combinatorial fact is what produces the low-dimensional exceptional sets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The exceptional-set bounds are qualitative; extending the determinant estimates to families of maps with parameters would likely give quantitative dimension bounds for bad pins or vantage points, with the $(d-2)$-flat bound for pins a natural sharpness target.
  • The same canonical-embedding recipe should apply to other curve families with computable Jacobians, such as parabolas or hyperbolas, producing new 'good curve' theorems for 1-rectifiable sets.
  • The localization gap in the proofs suggests a standalone lemma—every 1-rectifiable set of positive length contains a bounded positive-measure subset on which a nondegenerate $C^1$ embedding is bilipschitz—that would complete the 'without loss of generality' steps and might generalize the framework beyond the three examples.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The manuscript develops a transfer principle for Federer's projection theorem: given a family of nonlinear maps φ_α: Ω→R whose canonical embedding H_α=(φ_{α1},...,φ_{αd}) is locally bilipschitz or has Jacobian bounded below, one applies Federer's theorem to H_α(E) to conclude that at least one nonlinear image φ_{α_j}(E) has positive length. This framework is applied to three problems: pinned distance sets for 1-rectifiable sets not essentially (d−1)-flat (Theorem 2.3) with structural consequences for the set of bad pins (Theorem 2.6); radial projections in the plane (Theorem 3.1); and unions of variable-radius circles centered on a 1-rectifiable set (Theorem 4.2). The paper also proves a short d-lines corollary to illustrate the method.

Significance. The framework is attractive and, if correct, would provide a genuinely common route to several results at the rectifiable endpoint, including sharp statements on exceptional sets. The Jacobian computations in Lemmas 2.4 and 3.2 are explicit and check out, and the use of Federer's theorem as a black box makes the combinatorial step in Sections 2 and 3 conceptually clean. However, the proof of Theorem 4.2 rests on a determinant estimate in Lemma 4.3 that appears to be false as stated, and the proofs of Theorems 2.3, 3.1, and 4.2 contain an unstated bounded-subset reduction. These issues are local and fixable in principle, but they are load-bearing for the central claims.

major comments (2)
  1. [Section 2, Proof of Theorem 2.3; also Section 3 and Section 4] The inference from H^1(E \ V_P(ε))>0 to "we may therefore assume without loss of generality that ... E⊆Ω_ε" is logically inverted and skips a necessary boundedness reduction. Having H^1(E \ V_P(ε))>0 only means that a positive-measure part of E lies at distance at least ε from the affine hull, not that all of E lies in Ω_ε=R^{d-1}×(ε,1/ε); moreover Ω_ε is unbounded in the first d−1 coordinates. Since Lemma 2.4 only bounds det DH_P on Ω_ε and the columns of DH_P are 2(z−p_j), a determinant bound does not control the operator norm of DH_P on unbounded sets, and the Jacobian lower bound alone does not imply H^1(H_P(E))>0 for unbounded E. The proof should first choose a bounded positive-measure Borel subset E'⊆E∩V_P(ε)^c, reflect or rotate so that E'⊆Ω_ε∩B_R, apply the area formula to H_P(E'), and then note that a good pin for E' is a good pin for E because ∆_p(E')⊆∆_p(E). The same missing reduction appears in the proof of Theorem 3.1 (assumption E⊆Ω_ε) and in the proof of Theorem 4.2 (assumption E⊆(a/10,11a/100)×R).
  2. [Lemma 4.3, determinant decomposition] In the decomposition det DH_(α,β) = (A)+(B)+(C), term (C) has the wrong sign: since α<β and r_y≥0, the expression (C) = (r_y/r)·(α−β)/(√(1−((x−α)/r)^2)√(1−((x−β)/r)^2)) is nonpositive, not nonnegative as claimed. The lower bound for det DH therefore depends on whether |C| can dominate the positive contribution (A) ≥ (α−β)/∥r∥∞. It can: with r(x,y)=a+(b−a)/(1+e^{−Ky}) (so r_x=0, r_y≥0, bounded range and bounded gradient), term (C) is approximately −K(b−a)(α−β)/(4r), which for large K exceeds the positive lower bound (α−β)/b. Thus Lemma 4.3's assertion that the Jacobian determinant is bounded away from zero is false without an additional smallness condition on ∥∇r∥∞, and the proof of Theorem 4.2 collapses at this point. A corrected estimate or a modified hypothesis on r is required before the circle-union theorem can be accepted.
minor comments (5)
  1. [Section 1.2] The word "bilipshictz" should be "bilipschitz", and "1-unrectifable" should be "1-unrectifiable".
  2. [Proof of Corollary 1.3] The notation π_j(t)∩H(E) should be π_j^{-1}(t)∩H(E) in two places, since π_j maps R^d to R and H(E) is a subset of R^d.
  3. [Proof of Theorem 2.3] The sentence claiming that H_P is "locally Lipschitz due to the upper bound on detDH_P" is imprecise: local Lipschitzness follows from boundedness of the full derivative matrix on bounded subsets, not from a determinant upper bound alone.
  4. [Remark 2.2 and Section 3] The definition of essentially k-flat in Remark 2.2 differs from the Orponen–Sahlsten definition, and Section 3 later refers to "k-flat" without repeating the variant; this distinction should be stated where the term is used in Theorem 3.1.
  5. [Proof of Theorem 4.2] The final integration uses |φ_γ(E)|>0 for all γ in an interval, but the argument before it only shows this for γ in a set of positive measure or for a dichotomy of intervals; the Fubini step should be written more carefully to make clear how pointwise positivity on a full positive-length interval is obtained.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the nonlinear projection framework is an honest transfer of Federer's theorem; self-citations are comparative, not load-bearing.

full rationale

The derivation chain is self-contained with respect to its stated external input, Federer's projection theorem. The canonical embeddings H_P, H_±, and H_(α,β) are constructed from the relevant nonlinear maps (squared distance, arctangent angles, circle-intersection ordinates), and none of these maps is defined in terms of the conclusions being proved (e.g., a 'good pin' or positive area). Positivity of H^1(H(E)) is intended to follow from the Jacobian/bilipschitz estimates in Lemmas 2.4, 3.2, and 4.3, followed by Federer's theorem applied to H(E); this is a genuine transfer rather than a renaming of the conclusion. The self-citations ([9], [11], [12], [14]) are disclosed as comparisons, alternative proofs, or future work and are not load-bearing inputs; in particular, [12] is noted only after the paper's own proofs are completed. There are no fitted parameters, and no exceptional-set statement is assumed in order to prove itself. The genuine weaknesses are proof gaps rather than circularity: the WLOG reductions in the proofs of Theorems 2.3, 3.1, and 4.2 ('E⊆Ω_ε' and 'E⊆(a/10,11a/100)×R') are asserted without the bounded-subset or localization argument that would justify them, and a determinant lower bound alone does not imply positivity of the image for unbounded E. These gaps would require an additional argument, but none of them exhibits a step where the theorem's output is secretly an input. Hence no significant circularity is present, and the score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities: the paper is a theorem/proof contribution. The axioms are standard theorems in geometric measure theory plus the stated regularity conditions on E and r.

assumptions (6)
  • standard math Federer's projection theorem (Theorem 1.1 in the paper): H^1(E) is comparable to the sum of multiplicity-weighted projection lengths onto an orthonormal basis.
    The central black box used in every application; it is cited from Federer [7, Theorem 3.2.27].
  • standard math Besicovitch-Federer projection theorem (Theorem 1.2)
    Used for the converse direction in Corollary 1.3 and to characterize purely 1-unrectifiable sets.
  • standard math Local bilipschitz equivalence follows from a Jacobian determinant bounded away from zero and infinity (inverse function theorem / area formula)
    Invoked in Lemmas 2.4, 3.2, 4.3 to transfer rectifiability and positivity of measure to H(E).
  • standard math Hausdorff measure is Borel regular and sigma-finite, so a positive-measure set contains a bounded positive-measure subset
    Needed to repair the 'E ⊆ Ω_ε' reductions in Theorems 2.3 and 3.1; not stated in the paper but standard.
  • standard math Fubini-Tonelli theorem
    Used at the end of Theorem 4.2 to convert vertical-fiber positivity into area positivity.
  • domain assumption The radius function r satisfies 0<a≤r≤b, is differentiable with bounded gradient, and its partial derivatives do not change sign (Definition 4.1)
    This admissibility condition is what makes the Jacobian lower bound in Lemma 4.3 hold; Talagrand's example shows some condition is necessary.

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Cite this review

Pith. "Pith review of Applications of Nonlinear Projections to Rectifiable 1-sets." pith.science (2026). https://pith.science/paper/Z4VIMEFM

@misc{pith2026260810253,
  author       = {Pith},
  title        = {Pith review of: Applications of Nonlinear Projections to Rectifiable 1-sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z4VIMEFM}},
  note         = {Machine review of arXiv:2608.10253}
}
abstract

Projection theorems in Euclidean space provide a fundamental link between the geometric structure of a set and the size of its lower-dimensional images. For 1-rectifiable sets in $\mathbb{R}^d$, a classical theorem of Federer shows that the 1-dimensional Hausdorff measure of such sets is controlled by the multiplicity-weighted lengths of finitely many linearly independent projections. We develop a framework for extending Federer's result into a diverse set of nonlinear problems. This technique yields a unified approach for studying sets through their lower-dimensional nonlinear images, as well as studying the exceptional sets which exhibit poor projective behavior. As illustrations of our technique, we show that (i) every 1-rectifiable set contains a pin whose pinned distance set has positive Lebesgue measure, and that the exceptional set of pins for which this fails is contained in a $(d-2)$-dimensional affine subspace; (ii) planar radial projections of a 1-rectifiable set can fail to have positive length from at most one vantage point unless the set is essentially linear; and finally (iii) unions of circles centered on a 1-rectifiable set have positive area under mild assumptions on the radius function.

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Reference graph

Works this paper leans on

29 extracted references · 21 canonical work pages

  1. [1]

    Bourgain

    J. Bourgain. Averages in the plane over convex curves and maximal operators.Journal d’Analyse Math´ ematique, 47:69–85, 1986

    work page 1986

  2. [2]

    Bright, Y

    P. Bright, Y. Fu, and K. Ren. Radial projections inR n revisited.arXiv preprint, arxiv.org/abs/2406.09707, 2024. 16 R. BONGERS, P. BRIGHT, C. MARSHALL, AND K. TAYLOR

    arXiv 2024

  3. [3]

    Bright and S

    P. Bright and S. Gan. Exceptional set estimates for radial projections inR n.Annales Fennici Mathe- matici, 49(2):631–661, 2024

    work page 2024

  4. [4]

    Cs¨ ornyei and D

    M. Cs¨ ornyei and D. M. Stull. Improved bounds for radial projections in the plane.arXiv preprint, arxiv.org/abs/2508.18228, 2025

    arXiv 2025

  5. [5]

    X. Du, Y. Ou, K. Ren, and R. Zhang. New improvement to Falconer distance set problem in higher dimensions.arXiv preprint, arXiv:2309.04103, 2023

    arXiv 2023

  6. [6]

    K. J. Falconer. On the Hausdorff dimensions of distance sets.Mathematika, 32(2):206–212, 1985

    work page 1985

  7. [7]

    Federer.Geometric Measure Theory

    H. Federer.Geometric Measure Theory. Classics in Mathematics. Springer, 1996

    work page 1996

  8. [8]

    L. Guth, A. Iosevich, Y. Ou, and H. Wang. On Falconer’s distance set problem in the plane.Invent. Math., 219(3):779–830, 2020

    work page 2020

Show all 29 references
  1. [9]

    Iosevich, Z

    A. Iosevich, Z. Li, and K. Taylor. Positive measure of unions of variable surfaces.arXiv preprint, arXiv:2605.27550, 2026

    2026 arXiv

  2. [10]

    I. Laba. Recent progress on Favard length estimates for planar Cantor sets. InOperator-related function theory and time-frequency analysis, volume 9 ofAbel Symp., pages 117–145. Springer, Cham, 2015

    2015

  3. [11]

    I. Laba, A. McDonald, and K. Taylor. Favard length and generalized projections.arXiv preprint, arXiv:2607.28793, 2026

    2026 arXiv

  4. [12]

    Li and K

    Z. Li and K. Taylor. A quantified two-projection theorem for nonlinear projections.arXiv preprint, arxiv.org/abs/2606.00381, 2026

    2026 arXiv

  5. [13]

    Marshall

    C. Marshall. Power laws for the Favard length problem inR d.arXiv preprint, arxiv.org/abs/2509.02882, 2025

    2025 arXiv

  6. [14]

    Marshall

    C. Marshall. Finite Good Witnesses for Generalized Curve Projections at the Rectifiable Endpoint. Preprint, to appear on arXiv, 2026

    2026

  7. [15]

    J. M. Marstrand. Some fundamental geometrical properties of plane sets of fractional dimensions.Proc. London Math. Soc. (3), 4:257–302, 1954

    1954

  8. [16]

    J. M. Marstrand. Packing circles in the plane.Proceedings of the London Mathematical Society, 55:37–58, 1987

    1987

  9. [17]

    Mattila.Geometry of sets and measures in Euclidean spaces, volume 44 ofCambridge Studies in Advanced Mathematics

    P. Mattila.Geometry of sets and measures in Euclidean spaces, volume 44 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995. Fractals and rectifiability

    1995

  10. [18]

    Mattila and T

    P. Mattila and T. Orponen. Hausdorff dimension, intersections of projections and exceptional plane sections.Proceedings of the American Mathematical Society, 144(8):3419–3430, 2016

    2016

  11. [19]

    T. Mitsis. On a problem related to sphere and circle packing.Journal of the London Mathematical Society, 60(2):501–516, 1999

    1999

  12. [20]

    Nazarov, Y

    F. Nazarov, Y. Peres, and A. Volberg. The power law for the Buffon needle probability of the four-corner Cantor set.Algebra i Analiz, 22(1):82–97, 2010

    2010

  13. [21]

    D. M. Oberlin. Unions of hyperplanes, unions of spheres, and some related estimates.Illinois Journal of Mathematics, 51(4):1265–1274, 2007

    2007

  14. [22]

    T. Orponen. A sharp exceptional set estimate for visibility.Bulletin of the London Mathematical Society, 50(1):1–6, 2017

    2017

  15. [23]

    T. Orponen. On the dimension and smoothness of radial projections.Anal. PDE, 12(5):1273–1294, 2019

    2019

  16. [24]

    Orponen and T

    T. Orponen and T. Sahlsten. Radial projections of rectifiable sets.Ann. Acad. Sci. Fenn. Math., 36(2):677–681, 2011

    2011

  17. [25]

    Orponen, P

    T. Orponen, P. Shmerkin, and H. Wang. Kaufman and Falconer estimates for radial projections and a continuum version of Beck’s theorem.GAF A Geom. Funct. Anal., 34(1):164–201, 2024

    2024

  18. [26]

    K. Ren. Discretized radial projections inR d.arXiv preprint, arxiv.org/abs/2309.04097, 2025

    2025 arXiv

  19. [27]

    Simon and K

    K. Simon and K. Taylor. Dimension and measure of sums of planar sets and curves.Mathematika, 68(4):1364–1392, 2022

    2022

  20. [28]

    Talagrand

    M. Talagrand. Sur la mesure de la projection d’un compact et certaines familles de cercles.Bull. Sci. Math. (2), 104(3):225–231, 1980

    1980

  21. [29]

    T. Wolff. Local smoothing type estimates onL p for largep.Geometric and Functional Analysis, 10(5):1237–1288, 2000. APPLICATIONS OF NONLINEAR PROJECTIONS 17 Department of Applied Mathematics, University of California, Merced, Merced, CA Email address:rosemariebongers@ucmerced....

    2000

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