Pinned Dot Product Set Estimates

Caleb Marshall; Paige Bright; Steven Senger

REVIEW 2 major objections 6 minor 1 cited by

Pinned Dot Product Set Estimates

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Pith's one-line read For fractals in R^n, pinned dot product sets become large once the dimension exceeds n/2, the same threshold conjectured for Falconer distances.

desk verdict Nice projection framework, but the advertised n/2 threshold for pinned dot products is not proved: Theorem 1.6 lacks the key full-dimensionality step for the pin set. read the letter →

arxiv 2412.17985 v1 pith:I2JEY4SS submitted 2024-12-23 math.CA math.MG

Paige Bright , Caleb Marshall , Steven Senger This is my paper · ORCID

classification math.CAmath.MG MSC 28A8028A75

keywords pinneddotproductsFalconerdistanceproblemHausdorffdimensionradialprojectionsorthogonalexceptionalsetsk-planardispersiongeometricmeasuretheory

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

This paper studies how large a fractal set A in R^n must be so that its pinned dot product sets are large. For a fixed translation x, the threshold is (n+1)/2: whenever dim A > (n+1)/2, a full-dimensional subset of pins a in A has Π^a_x(A) with positive 1-dimensional Hausdorff measure, and analogous thresholds give high dimension or nonempty interior. The main new result is translation-dependent: if the set of allowed translations X is k-planar dispersed and dim X ≥ dim A, then dim A > n/2 already forces positive-measure dot product sets for a full-dimensional family of translations and pins. This matches the conjectured Falconer distance threshold, supporting the authors' conjecture that for any Borel A with dim A > n/2 there is a pin a with $H^{1}$(Π^a_0(A)) > 0.

What carries the argument

The load-bearing identity is Lemma 2.9: for a ≠ x, H^s(Π^a_x(A)) = c_{a,x,s} H^s(P_{π_x(a)}(A)), where P_θ is orthogonal projection onto the line spanned by θ and π_x(a) = (a-x)/|a-x| is the radial projection of the pin onto the unit sphere. This converts dot product sets into projection theory: one needs directions π_x(a) that avoid the exceptional sets E^+(A), E^u(A), E^◦(A). The argument combines classical exceptional set estimates (Prop 2.3), the elementary radial projection bound dim π_x(A) ≥ dim A - 1, and, for the strengthened theorem, Ren's discretized radial projection theorem (Theorem 2.7) in the form of Corollary 2.8, which guarantees many translations x in X with large radial projection dimension when X has k-planar dispersion.

What would settle it

A concrete refutation would be a Borel set A ⊂ R^n with dim A > n/2 and a k-planar dispersed set X with dim X ≥ dim A such that, for every full-dimensional subset of X, the set {x ∈ X : dim π_x(A) ≤ n - dim A} already has full dimension in X; that would contradict Corollary 2.8 and collapse Theorem 1.6.(2).

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Extended reading notes

Core claim

The paper's central discovery is that the pinned dot product problem reduces, through the identity $H^{1}$(Π^a_x(A)) = c $H^{1}$(P_{π_x(a)}(A)), to comparing the radial projection set π_x(A) with the exceptional set of orthogonal projections that shrink A. Classical exceptional set bounds give the translation-invariant threshold (n+1)/2. The new input is Corollary 2.8: for k-planar dispersed X, except for a dimension-negligible subset of x in X, dim π_x(A) ≥ min{dim X, dim A, k}; taking k = ceil(n/2), this beats the exceptional set dimension whenever dim A > n/2, yielding Theorem 1.6.(2). The proof then produces full-dimensional sets X_A ⊂ X and A_x ⊂ A for which every pinned dot product set has positive measure. A corollary gives a full-dimensional set of difference vectors a0 in A - A with $H^{1}$(a0 · A) > 0.

Load-bearing premise

The whole translation-dependent improvement rests on Ren's discretized radial projection theorem, taken as a black box: if its stated conclusion does not hold for general Borel A and X, or if the epsilon loss in the derived Corollary 2.8 cannot be absorbed into the strict dimension gap, the n/2 threshold falls. The classical bound dim E^+(A) ≤ n - dim A is also assumed without proof.

Editorial extensions

If this is right

If Theorem 1.6.(2) is correct, the pins/translations version of the dot product Falconer problem is solved at the conjectured threshold n/2, the same as the distance problem.
Corollary 1.9 gives a full-dimensional set of difference vectors a0 in A - A with H^1(a0 · A) > 0, extending the dot product result to differences.
For even n, any set with dim A > n/2 is automatically k-planar dispersed, so the main theorem covers all even-dimensional sets at the threshold.
The sharper planar bound of Ren and Wang on exceptional sets would further lower the dimensional thresholds in Theorem 1.2.(1) and Theorem 1.6.(1) for n = 2, as noted in Remark 2.4.
The tree application (Theorem 4.1) extends the pinned dot product result to k-trees of dot products: for dim A > (n+1)/2, there is a full-dimensional set of roots with positive k-dimensional measure of edge-weight tuples.

Reading between the lines

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

The same projection framework likely applies to other pinned configurations (e.g., pinned volumes or angles) where the configuration map is a submersion onto hyperplanes, so improvements in radial projection theorems should transfer directly.
A natural test of the k-planar dispersion hypothesis: construct a Borel set X with dim X ≥ s but without k-planar dispersion for which the conclusion of Theorem 1.6 fails; if such a set exists, the dispersion assumption would be shown necessary.
Because the epsilon loss in Corollary 2.8 is absorbed only by strict inequality s > n/2, the theorem does not address the critical case s = n/2; one might expect the conclusion to fail there, analogously to the distance problem.
The sharpness example A = C × {0} in the plane shows the n/2 threshold cannot be lowered in general, so the theorem is optimal in dimension without additional Fourier or structural hypotheses.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the short version: the paper has a clean and genuinely useful framework, but the main advertised result, Theorem 1.6, has a load-bearing gap. The proof never shows that the pin set A_x is full-dimensional in A.

What is actually new and good: the reduction of pinned dot product sets to orthogonal projections is elegant and correct. Lemma 2.9 is an exact scaling identity, and the discussion connecting exceptional sets for projections to dot product size is a nice unifying perspective. The new parts of Theorem 1.2 (parts (1) and (3)) look solid and are genuinely new; part (2) correctly re-proves the known Iosevich–Taylor–Uriarte-Tuero result with classical tools. The translation-dependent formulation with k-planar dispersion and the use of Ren's and Orponen–Shmerkin–Wang's radial projection theorems is a sensible and interesting route, and the survey of the state of the art is a useful service.

The soft spot is not a typo. In the proof of Theorem 1.6.(2), the set of directions A_x = π_x(A) \ E_+(A) is shown to be large in π_x(A). But the pin set is the preimage of that set inside A, and no argument shows this preimage has dimension equal to dim A. For that you need to bound the bad set A \ A_x, and the only available argument (the one used in Theorem 1.2) gives dim(A \ A_x) ≤ dim E_+(A) + 1 ≤ n−s+1. That is smaller than s only when s > (n+1)/2, which is precisely Theorem 1.2's range, not the advertised improvement. So in the range n/2 < s ≤ (n+1)/2, the main conclusion is unproved. The stress-test scenario is plausible and worth taking seriously: a lower-dimensional good subset can lift the radial projection above n−s while the good pins stay low-dimensional. Corollary 1.9 inherits this gap. The same issue affects parts (1) and (3) of Theorem 1.6.

Minor issues: the notation A_x is used for both the direction set and the pin set; the proof of Theorem 1.6 says k = floor((n+1)/2) while the statement has k = ceil(n/2) (these coincide, so it is not a real conflict); and the tree construction in Section 4 is only sketched.

Who should read this: anyone working on configuration problems or projection theory will get something from the framework and the survey. The new parts of Theorem 1.2 are worth citing once they are vetted. But do not cite Theorem 1.6 or Corollary 1.9 in their present form.

Recommendation: this deserves a serious referee, but the referee should be sent to work on the full-dimensionality step. Either the authors prove it under additional hypotheses on A (e.g., k-planar dispersion of A) or they soften the statement. As is, the main threshold improvement is not established.

Referee Report

2 major / 6 minor

Summary. The paper studies pinned dot product sets \Pi^a_x(A):=\{(a-x)\cdot y:y\in A\} for Borel sets A\subset\mathbb{R}^n and parameters a,x\in\mathbb{R}^n. It proves translation-invariant results (Theorem 1.2) that give dimensional thresholds for \Pi^a_x(A) to have Hausdorff dimension at least u, positive H^1 measure, or nonempty interior, using classical exceptional-set estimates for orthogonal projections. It then states translation-dependent results (Theorem 1.6) that lower these thresholds to the Falconer-type exponent n/2 under a k-planar dispersion assumption on a set X of translations, via recent radial projection theorems of Orponen\textendash Shmerkin\textendash Wang and Ren. The paper also contains a Fourier-dimension result, a restricted-projection result using Zahl's theorem, and applications to dot-product trees and sharpness constructions. The Key Lemma 2.9, which identifies the size of \Pi^a_x(A) with the size of the orthogonal projection P_{\pi_x(a)}(A), is a direct and elegant reduction.

Significance. If the full statement of Theorem 1.6 were established, it would give a pinned dot product result at the same dimensional threshold as the Falconer distance conjecture, which is an attractive and significant outcome. The paper's projection-theoretic framework is clean, and the translation-invariant Theorem 1.2 appears to be correctly proved and provides a useful unified treatment. Proposition 1.4 and Proposition 1.10 are also nice consequences of known projection results. However, the proof of Theorem 1.6 contains a substantial gap: it never proves that the pin set A_x is full-dimensional in the advertised range n/2<s\le (n+1)/2, and the proof of Corollary 1.9 additionally fails to justify the existence of a single full-dimensional pin set common to all translates. These gaps directly affect the paper's main new claims.

major comments (2)

[Section 3.2, proof of Theorem 1.6(2)] The proof defines A_x=\pi_x(A)\setminus E_+(A) and then A_x=\pi_x^{-1}(A_x)\cap(A\setminus\{x\}), and verifies that H^1(\Pi^a_x(A))>0 for every a in the second set. It never proves that this A_x has dim A_x=\dim A, which is part of the theorem's conclusion. For A^*=A\setminus A_x=\{a\in A:\pi_x(a)\in E_+(A)\}, the only bound available is \dim A^*\le \dim E_+(A)+1\le n-s+1. This gives \dim A^*<s only when s>(n+1)/2, which is exactly the range already covered by Theorem 1.2(2). In the advertised improvement range n/2<s\le(n+1)/2 the argument does not exclude the possibility that the bad complement has full dimension. The analogous gap affects Theorem 1.6(1), where the corresponding bound would require s>(n+u)/2, and Theorem 1.6(3), where it would require s>(n+2)/2. Thus the central claim that Theorem 1.6 improves the threshold to n/2 is not established by the given proof.
[Corollary 1.9, proof] The proof takes X=A in Theorem 1.6 and obtains a full-dimensional set A_1\subset A and, for each x\in A_1, a full-dimensional set A_x\subset A with H^1(\Pi^a_x(A))>0 for all a\in A_x. The conclusion (1.3), however, requires a single full-dimensional set A_2 such that positivity holds for every (a_1,a_2)\in A_1\times A_2; this would require A_2\subset\bigcap_{a_1\in A_1}A_{a_1}. The proof simply renames the pointwise sets A_x as A_2, without any argument that a common full-dimensional intersection exists. This does not follow from Theorem 1.6 as stated, so the advertised consequence that there is a full-dimensional set A_0\subset A-A with H^1(\Pi^{a_0}_0(A))>0 is unsupported.

minor comments (6)

[Sections 3.1 and 3.2] The symbol A_x is used both for the set of directions \pi_x(A)\setminus E(A) and for the pin set \pi_x^{-1}(A_x)\cap A. This overloading makes the proofs difficult to follow; using distinct notation such as D_x and A_x would improve clarity.
[Proof of Theorem 1.6] The proof sets k=\lfloor(n+1)/2\rfloor while the statement uses k=\lceil n/2\rceil. The two expressions are equal for integer n, but the inconsistency should be resolved in a revision.
[Proof of Theorem 1.2(2), Section 3.1] The line 'A^*\subset E_+(A)' is a type error: A^* is a subset of \mathbb{R}^n while E_+(A) is a subset of S^{n-1}. The intended statement is \pi_x(A^*)\subset E_+(A).
[Proof of Proposition 2.6] The inclusion A\subset\pi_x(A\setminus\{x\})\times\hat{A}_x is informal; the polar-coordinate map (\theta,r)\mapsto x+r\theta should be stated explicitly so that the dimension bound is rigorously justified.
[Theorem 4.1 statement] The statement repeats 'for all x\in A_v' and does not assert any lower bound on \dim A_v; as written, the result is close to vacuous. The intended quantitative assertion should be stated.
[Application of Corollary 2.8 in Theorem 1.6] The proof does not explicitly specify the choice of \epsilon in Corollary 2.8. Because s>n/2 gives \min\{\dim X,\dim A,k\}-(n-s)\ge s-n/2>0, a sufficiently small \epsilon exists, but this should be spelled out.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main theorems are deduced from imported projection theorems and an explicit scaling identity; author self-citations are not load-bearing.

full rationale

No circular step is present. The central reduction (Key Lemma 2.9) is a literal scaling identity: H^s(Π^a_x(A)) = |a-x|^s H^s(P_{π_x(a)}(A)), so it equates the dot-product set to an orthogonal projection without importing the conclusion. Theorem 1.2 is proved from classical exceptional-set estimates (Kaufman, Mattila, Falconer, Peres-Schlag) and the elementary bound dim π_x(A) ≥ dim A - 1, for which a self-contained covering proof is given. Theorem 1.6 is derived from Ren's external discretized radial projection theorem via Corollary 2.8; k is chosen from the dimension threshold and X_A is defined by exactly the inequality that Corollary 2.8 supplies, so no parameter is fitted to the desired conclusion. The few author self-citations (BFR24, KMS23, IS20, IS16, Aut+24) occur in peripheral roles: an alternative proof of Proposition 2.6 and sharpness/tree constructions in Section 4; they are not the premise for the main new theorem. One non-circular gap should be flagged: the proof of Theorem 1.6 never establishes that the pin set A_x is full-dimensional in A; it only shows the radial direction set π_x(A)\E_+(A) has dimension ≥ ψ_n(A,X) and then defines A_x as its preimage. This is a missing argument about dim(π_x^{-1}(A_x)∩A), not a circularity; it is a correctness concern for the advertised range s>n/2 and does not affect the circularity score.

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

The central claims are proven by reducing dot product sets to orthogonal projections and importing known projection theorems. The free parameter ledger is empty because the theorems are universal statements with no fitted constants. The main axioms are the external projection and radial projection results, especially Ren's discretized theorem and Zahl's restricted projection theorem, plus the standard exceptional set estimates.

assumptions (6)

standard math Exceptional set estimates for orthogonal projections (Prop 2.3): dim E_u(A) <= max(n-1+u-dim A, 0); dim E_+(A) <= n-dim A if dim A>1; dim E_o(A) <= n+1-dim A if n>=3 and dim A>2.
Used as black boxes from Kaufman, Mattila, Falconer, Peres-Schlag via [Mat15, Ch.5]; load-bearing for all parts of Theorem 1.2 and Theorem 1.6.
standard math Radial projection dimension bound dim π_x(A) >= dim A - 1 for Borel A with dim A > 1 (Prop 2.6).
Used to ensure the radial image of A has larger dimension than the exceptional set; proof is sketched in the paper.
standard math Ren's discretized radial projection theorem (Thm 2.7): sup_{x in X} dim π_x(A) >= min(dim X, dim A, k) when X is not contained in a k-plane.
External deep result [Ren23, Thm 1.1]; the engine for the improved thresholds in Theorem 1.6.
standard math Zahl's restricted projection theorem (Thm 3.1): for any non-degenerate curve γ, dim{t in [0,1] : dim P_{γ(t)}(A) < u} <= u.
External result [Zah23, Thm 1.10], with the planar case due to Pramanik-Yang-Zahl; used for Proposition 1.10.
standard math Fourier dimension is at most Hausdorff dimension (Prop 1.3).
Cited from [Mat99, Sec 12.17]; used in Proposition 1.4.
domain assumption The k-planar dispersion condition (Definition 1.5) is well-defined and the derivation of Corollary 2.8 from Theorem 2.7 is correct for sets satisfying it.
This hypothesis on X is what makes the translation-dependent improvements possible; if it is too restrictive or the corollary fails, Theorem 1.6 loses its force.

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Pith. "Pith review of Pinned Dot Product Set Estimates." pith.science (2026). https://pith.science/paper/I2JEY4SS

@misc{pith2026241217985,
  author       = {Pith},
  title        = {Pith review of: Pinned Dot Product Set Estimates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I2JEY4SS}},
  note         = {Machine review of arXiv:2412.17985}
}

abstract

We study a variant of the Falconer distance problem for dot products. In particular, for fractal subsets $A\subset \mathbb{R}^n$ and $a,x\in \mathbb{R}^n$, we study sets of the form \[ \Pi_x^a(A) := \{\alpha \in \mathbb{R} : (a-x)\cdot y= \alpha, \text{ for some $y\in A$}\}. \] We discuss some of what is already known to give a picture of the current state of the art, as well as prove some new results and special cases. We obtain lower bounds on the Hausdorff dimension of $A$ to guarantee that $\Pi^a_x(A)$ is large in some quantitative sense for some $a\in A$ (i.e. $\Pi_x^a(A)$ has large Hausdorff dimension, positive measure, or nonempty interior). Our approach to all three senses of "size" is the same, and we make use of both classical and recent results on projection theory.

Figures

Figures reproduced from arXiv: 2412.17985 by the authors.

**Figure 1.** Two hyperplanes corresponding to elements α1, α2 ∈ Πa x (A). Utilizing the exceptional set E(A), the pinned dot product problem turns into the following question: is there a direction in the set of possible directions to orthogonally project onto, πx(A), that does not lie in the set of bad directions, E(A)? Using standard results on dim πx(A) and dim E(A), we obtain that there are in fact many directions in πx(A) no… view at source ↗

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Forward citations

Cited by 1 Pith paper

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A Structural Condition on Point Sets with Few Distinct Dot Products
math.CO 2025-10 reject novelty 6.0 of 10

Any point set in the plane with o(n^{3/4}) distinct dot products must contain a line through the origin holding n^{1/2} points whose consecutive distance ratios cluster near 1.

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Works this paper leans on

11 extracted references · 4 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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[Du+23b] Xiumin Du, Yumeng Ou, Kevin Ren, and Ruixiang Zhang

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On Falconer’s distance set conjecture

arXiv: 2309.04501 [math.CA]. [Erd06] M. Burak Erdo˘ gan. “On Falconer’s distance set conjecture”. In: Revista Matem´ atica Iberoamericana 22.2 (2006), pp. 649–662. [Fal03] Kenneth Falconer. Fractal Geometry. John Wiley & Sons, Ltd,

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work page 1999

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Hausdorff dimension and the exceptional set of projec- tions

[Fal82] Kenneth J. Falconer. “Hausdorff dimension and the exceptional set of projec- tions”. In: Mathematika 29.1 (1982), pp. 109–115. [Fal85] K. J. Falconer. “On the Hausdorff dimensions of distance sets”. In: Mathematika 32.2 (1985), pp. 206–212. doi: 10.1112/S0025579300010998. [GIT19] Allan Greenleaf, Alex Iosevich, and Krystal Taylor. “Configuration S...

doi:10.1112/s0025579300010998 1982

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Hausdorff dimension, orthogonal projections and intersections with planes

[Mat75] Pertti Mattila. “Hausdorff dimension, orthogonal projections and intersections with planes”. In: Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), pp. 227–244. [Mat85] Pertti Mattila. “On the Hausdorff dimension and capacities of intersections”. In: Mathematika 32 (1985), pp. 213–217. [Mat87] Pertti Mattila. “Spherical averages of Fourier transforms ...

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