The Erd\H{o}s-Falconer distance problem between arbitrary sets and $k$-coordinatable sets in finite fields

Doowon Koh; Firdavs Rakhmonov; Hunseok Kang

REVIEW 1 major objections 5 minor 17 references

The Erd\H{o}s-Falconer distance problem between arbitrary sets and $k$-coordinatable sets in finite fields

T0 review · 1 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A k-coordinatable subset alone forces the distance set to have size at least q/2.

desk verdict Clean, correct proof that k-coordinatable sets get the sharp distance threshold; the structural restriction is explicit and the math holds up. read the letter →

arxiv 2506.07251 v1 pith:OLFCOHSV submitted 2025-06-08 math.CO math.NT

Hunseok Kang , Doowon Koh , Firdavs Rakhmonov This is my paper

classification math.COmath.NT MSC 52C1011T23

keywords finitefieldsFouriertransformdistanceslinesErdős-Falconerdistanceproblemk-coordinatableplanesL2restrictionestimatesBox

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 the number of distinct quadratic distances |Δ(A,B)| determined by two subsets A and B of F_q^d. Its main theorem states that when one of the sets lies in a k-coordinatable plane (a plane obtainable from a coordinate plane by rotation and translation), the distance set satisfies |Δ(A,B)| ≥ (1/2) min{ q, |A||B|/($2q^{{d-1}}$) }, so |A||B| ≥ 2q^d implies |Δ(A,B)| ≥ q/2. In the single-set case this recovers the sharp exponent (d+1)/2 for odd dimensions and, together with a constructed example, shows that no exponent below d/2 can hold in even dimensions. As an application, when 2 is a square in F_q the one-dimensional Box distance problem is improved from threshold $q^{{3/4}}$ to $q^{{2/3}}$.

What carries the argument

The central machine is an L2 restriction estimate for spheres applied to k-coordinatable sets. A k-coordinatable plane is an affine k-plane obtainable from a coordinate plane by a rotation and translation (equivalently, a plane whose direction subspace has a normal vector with square norm in F_q). Proposition 4.3 shows that for such a set B, max_t Σ_{m∈S_t} |\hat{B}(m)|² ≤ $2q^{{-d-1}}$|B|, where S_t = {x : ‖x‖ = t} is the standard sphere. This bound feeds into Lemma 4.1, a counting formula bounding the second moment of the number of pairs at distance t, which yields the distance-set lower bound. The proof of the restriction estimate works by rotating B to B_k × {0} and using the explicit sphere-size formula |S_t| ≤ $2q^{{d-1}}$ together with Plancherel.

What would settle it

Directly compute, for a small finite field F_q and a k-coordinatable set B such as a line L_λ with η(1+λ²)=1, the quantity max_{t∈F_q} Σ_{m∈$S_t^{{d-1}}$} |\hat{B}(m)|² and compare it to $2q^{{-d-1}}$|B|; exceeding this value would refute Proposition 4.3, which underpins the main theorem. Alternatively, search for sets A and such B with |A||B| ≥ 2q^d yet |Δ(A,B)| < q/2 in a computer search.

Watch

Extended reading notes

Core claim

The paper establishes that a single structural condition on one of the two sets is enough to force the distance set to be a positive fraction of the field: if A or B is k-coordinatable, then |Δ(A,B)| ≥ (1/2) min{ q, |A||B|/($2q^{{d-1}}$) }. In particular, the product condition |A||B| ≥ 2q^d gives |Δ(A,B)| ≥ q/2. The same result recovers the sharp (d+1)/2 exponent for the Erdős–Falconer distance problem in odd dimensions, and the companion construction Proposition 1.1 shows that no exponent smaller than d/2 is possible in even dimensions. As an application, the authors improve the Box distance problem in dimension one to threshold $q^{{2/3}}$ when 2 is a square.

Load-bearing premise

The theorem only applies when one of the two sets lies on a plane that a rotation can turn into a coordinate plane; because such a rotation must preserve the quadratic form, the needed square root must exist in the field, so planes like the line y=x in $F_q^{2}$ with η(2)=-1 are excluded.

Editorial extensions

If this is right

For any pair with |A||B| ≥ 2q^d where one set is k-coordinatable, the distance set is a positive proportion of the field.
In the single-set setting, any A containing a k-coordinatable subset of size |A|^α has |Δ(A)| ≥ q/2 once |A| ≥ 2^{1/(α+1)} q^{d/(1+α)}.
The Box-distance improvement for η(2)=1 shows a q^{2/3} threshold, beating the previous q^{3/4} for the same problem in dimension one.
The optimality construction (Proposition 1.1) settles that the (d+1)/2 exponent is best possible in odd dimensions, even without congruence restrictions on q.

Reading between the lines

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

The method suggests the real quantity controlling distances is the L2 restriction norm of B on spheres; any family of sets whose sphere-restriction energy is O(q^{-d-1}|B|) would inherit the same distance lower bound.
The condition η(2)=1 for the Box distance application may be an artifact of using the standard quadratic form; other equivalent quadratic forms could extend the improvement to fields where η(2)=-1.
Because rotations preserving the norm are exactly those whose direction subspace has a normal vector of square length, relaxing 'coordinatable' to a larger class of planes would require a genuinely different restriction estimate.

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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

This is a clean, honest paper. The main contribution is a generalized Erdős–Falconer bound when one of the two sets lies in a k-coordinatable plane: if |A||B| ≥ 2q^d then |Δ(A,B)| ≥ q/2. That recovers the sharp (d+1)/2 exponent for the classical Erdős–Falconer problem in odd dimensions and gives a new 2/3 exponent for the one-dimensional Box distance problem when 2 is a square.

The proof is mostly a pleasure to read. Lemma 4.1 is a sharpened version of the standard counting argument, using the explicit Fourier transform of the homogeneous variety V0; the step where the negative term is dropped is justified because that term is real and non-negative. The engine is Proposition 4.3, the L2 restriction estimate for k-coordinatable sets, and I checked the normalization: the q^{-d+k} factor is consistent with the Fourier normalization on F_q^k, so I don't see the notational slip mentioned in one review pass. The sharpness construction in Proposition 1.1 completes earlier counterexamples and is explicitly credited to [12]; the reproduction is fine.

The soft spots are proportional to what the theorem actually claims. The k-coordinatable hypothesis is genuinely restrictive. The WLOG reduction in Prop 4.3 only works when the plane is isometric to a coordinate k-plane via an orthogonal transformation preserving the quadratic form. For a totally isotropic 2-plane in F_q^4, the estimate fails, as the stress-test note correctly shows. But the paper does not overclaim: the scope is stated in Definition 1.5, and the y=x example in the introduction flags exactly why the condition is there. This is a scope limitation, not a hidden assumption. The two-dimensional Corollary 1.8 makes the friction visible: it only covers lines with η(1+λ^2)=1.

There are some typos (e.g., 'proeed') and the arXiv rendering of the contrast symbol is ugly, but nothing affects the math. The citation to [12] is a provenance note, not a circular dependency.

Who is this for? Anyone working on finite-field distance problems or extension estimates. It is a modest but real step: a structural condition that lowers the generalized threshold from q^{d+1} to q^d, with a short self-contained proof and a concrete application. I would be happy to referee it, and I'd accept after minor revisions. It deserves a serious referee, not a desk reject.

Referee Report

1 major / 5 minor

Summary. The paper studies the generalized Erdős-Falconer distance problem over finite fields, in which the distance set Δ(A,B) is determined by two sets A and B. The main result (Theorem 1.6) states that if at least one of A or B lies in a k-coordinatable plane — an affine k-plane that can be rotated and translated to a coordinate plane — then |A||B| ≥ 2q^d implies |Δ(A,B)| ≥ q/2. This is derived from a stronger quantitative bound (Theorem 5.1) via an L2 restriction estimate for spheres (Proposition 4.3). As applications, the paper recovers the sharp (d+1)/2 threshold for the single-set Erdős-Falconer distance problem in odd dimensions, proves the optimality of that exponent (Proposition 1.1), and improves the one-dimensional Box distance threshold from q^{3/4} to q^{2/3} when 2 is a square (Theorem 1.12).

Significance. If the proof is repaired as outlined below, the paper makes a solid contribution. It establishes a sharp threshold result for a natural class of structured sets, gives a self-contained proof of the optimality of the (d+1)/2 exponent in all odd dimensions and for all odd q, and provides a nontrivial application to the Box distance problem. The arguments are elementary and mostly transparent, and the constructions in Proposition 1.1 are explicit and verifiable. The restrictive k-coordinatable hypothesis is explicitly stated and honestly delimited, which is a strength rather than a hidden assumption.

major comments (1)

[Section 4.1, proof of Lemma 4.1] The displayed Fourier inversion identity in the proof of Lemma 4.1 is missing a minus sign. With the paper's normalization \hat{f}(m) = q^{-d}\sum_x f(x)\chi(-m\cdot x), Fourier inversion gives \sum_t \nu^2(t) = q^{4d}\sum_M \widehat{V0}(M)\widehat{A\times A}(-M)\widehat{B\times B}(M), not \sum_M \widehat{V0}(M)\widehat{A\times A}(M)\widehat{B\times B}(M). The subsequent substitutions and the expressions involving (\sum_{m\in S_t}\hat{A}(m)\hat{B}(m))^2 inherit the same sign issue. The final inequality (4.1) is nevertheless correct: replacing \widehat{A\times A}(M) by \widehat{A\times A}(-M) changes the square to (\sum_{m\in S_t}\hat{A}(-m)\hat{B}(m))^2, and the same Cauchy-Schwarz and non-negativity arguments apply. However, as written the proof of Lemma 4.1 is invalid, and since this lemma is the foundation of Theorem 5.1 and Theorem 1.6, the manuscript needs a corrected derivation.

minor comments (5)

[Definition 1.4] The notation for the coordinate axes contains a typo: "{ii, i2, ..., ik}" should be "{i_1, i_2, ..., i_k}".
[Proposition 4.3, proof] In the line "for m = (m_1,...,m_k,0) ∈ F_q^k × F_q^{d-k}", the notation \hat{0}(0) is undefined; it should be the Fourier transform of the indicator function of the singleton {0}, evaluated at the zero frequency.
[Proposition 1.1(ii), proof] The equality for the even-dimensional distance set has a typographical error: "|Δ_Q(A)| = {(a−b)^2 : a,b∈Ω_δ}|" should be "|Δ_Q(A)| = |{(a−b)^2 : a,b∈Ω_δ}|".
[Theorem 1.6 and Theorem 5.1] The statement of Theorem 1.6 (and Theorem 5.1) does not specify the range of k. The proof via Proposition 4.3 covers 1 ≤ k ≤ d−1; the case k=d is trivial and the case k=0 is vacuous under the size hypothesis |A||B| ≥ 2q^d. The authors should state this range explicitly or mention the trivial cases.
[Lemma 4.4, proof] There is a typo: "We now proeed to prove this" should be "We now proceed to prove this".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main bound is a direct Fourier-analytic derivation conditional on the explicit k-coordinatable hypothesis, and the optimality examples are proved in the paper.

full rationale

The paper's central claim, Theorem 1.6/5.1, is derived from Lemma 4.1 and Proposition 4.3. Lemma 4.1 is a counting/Cauchy-Schwarz identity proved in the text from the explicit Fourier transform of the homogeneous variety (Lemma 3.2, whose proof is included). Proposition 4.3 is the only place where the k-coordinatable hypothesis enters: the proof explicitly reduces B to B_k × {0} by rotation/translation, which is exactly what Definition 1.5 stipulates, and then bounds the restricted L2 sum using the sphere-size estimate (4.2) and Plancherel. The result is conditional on the stated hypothesis, not an equivalent reformulation of it. The restriction is genuine, as the paper's own y=x example and the skeptic's totally isotropic plane example show, but a scope limitation is not circularity. The sharpness construction (Proposition 1.1) is proved in Section 2 with a full construction and Lemma 2.2; the phrase 'follows the argument implicitly contained in [12]' is a provenance note, and the cited prior work is not used in place of a proof. No parameters are fitted and no quantity that is predicted is an input to the derivation. The application to the Box distance problem (Theorem 1.12) is a straightforward reduction to the main theorem under η(2)=1. Overall the derivation chain is self-contained against standard external lemmas (sphere sizes from [15], Fourier/Gauss sum facts proved or standard), so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central proof is self-contained Fourier analysis with no fitted parameters or invented entities. The only domain-specific postulate is the k-coordinatable plane condition, which is an explicit hypothesis of the theorem. External inputs are standard finite field results (Gauss sums, sphere sizes, quadratic form classification).

assumptions (5)

standard math Fourier analysis on finite abelian groups: orthogonality of characters, Fourier inversion, Plancherel (Section 3.1).
Used throughout Lemma 4.1 and Lemma 3.2; these are textbook facts for F_q^d.
standard math Gauss sum identities: G_1^2 = η(-1)q and the completing-square formula (3.1).
Invoked in the proof of Lemma 3.2 and in background; standard in finite field number theory.
standard math Sphere size estimates in Lemma 4.2 from Lidl-Niederreiter: max_t |S_t^{d-1}| ≤ 2q^{d-1}.
Used in Proposition 4.3 to bound the sphere average; cited to [15] and stated explicitly.
standard math Classification of non-degenerate quadratic forms over finite fields, equations (2.1)-(2.2).
Used only in the sharpness construction Proposition 1.1, not in the main theorem; cited to [1] and [7].
domain assumption The definition of k-coordinatable plane assumes that an affine k-plane can be mapped to a coordinate k-plane by an orthogonal transformation (rotation) over F_q preserving the standard quadratic form.
This is the structural restriction on the set B; without it Proposition 4.3 fails. The paper gives a two-dimensional example (line y=x needs η(2)=1).

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Cite this review

Pith. "Pith review of The Erd\H{o}s-Falconer distance problem between arbitrary sets and $k$-coordinatable sets in finite fields." pith.science (2026). https://pith.science/paper/OLFCOHSV

@misc{pith2026250607251,
  author       = {Pith},
  title        = {Pith review of: The Erd\Hos-Falconer distance problem between arbitrary sets and $k$-coordinatable sets in finite fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OLFCOHSV}},
  note         = {Machine review of arXiv:2506.07251}
}

abstract

In this paper, we study the cardinality of the distance set $\Delta(A, B)$ determined by two subsets $A$ and $B$ of the $d$-dimensional vector space over a finite field $\mathbb{F}_q$. Assuming that $A$ or $B$ lies in a $k$-coordinate plane up to translations and rotations, we prove that if $|A||B| > 2q^d$, then $|\Delta(A, B)| > q/2$, where $|\Delta(A, B)|$ denotes the number of distinct distances between elements of $A$ and $B$. In particular, we show that our result recovers the sharp $(d+1)/2$ threshold for the Erd\H{o}s-Falconer distance problem in odd dimensions, where distances are determined by a single set. As an application, we also obtain an improved result on the Box distance problem posed by Borges, Iosevich, and Ou, in the case where $2$ is a square in $\mathbb{F}_q$.

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Reference graph

Works this paper leans on

17 extracted references · 16 canonical work pages

[12]

Iosevich, D

A. Iosevich, D. Koh, and F. Rakhmonov, The quotient set of the quadratic distance set over ﬁnite ﬁel ds, Forum Math. 36 (2024), no. 5, 1341-1358

work page 2024
[1]

Ahmadi and A

O. Ahmadi and A. Mohammadian, Sets with many pairs of orthogonal vectors over ﬁnite ﬁelds , Finite Fields Appl. 37 (2016), 179-192

work page 2016
[2]

Bennett, D

M. Bennett, D. Hart, A. Iosevich, J. Pakianathan, and M. Ru dnev, Group actions and geometric combinatorics in Fd q, Forum Math. 29(2017), no. 1, 91-110

work page 2017
[3]

Borges, A

T. Borges, A. Iosevich, and Y. Ou, A singular variant of the Falconer distance problem , preprint (2023), arXiv:2306.05247

arXiv 2023
[4]

Bourgain, N

J. Bourgain, N. Katz, and T. Tao, A sum-product estimate in ﬁnite ﬁelds, and applications , Geometric & Func- tional Analysis, 14 (2004), no. 2, 27-57

work page 2004
[5]

Chapman, M

J. Chapman, M. Burak Erdo˘ gan, D. Hart, A. Iosevich, and D. Koh, Pinned distance sets, k-simplices, Wolﬀ ’s exponent in ﬁnite ﬁelds and sum-product estimates , Math Z. 271 (2012), no. 1, 63-93

work page 2012
[6]

Cheong, D

D. Cheong, D. Koh, and T. Pham, An asymmetric bound for sum of distance sets , Tr. Mat. Inst. Steklova 314 (2021), 290-300

work page 2021
[7]

Grove, Classical Groups and Geometric Algebra , Grad

L.C. Grove, Classical Groups and Geometric Algebra , Grad. Stud. Math. vol.39, American Mathematical Society, Providence, RI, 2002. 12

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[8]

Hanson, B

B. Hanson, B. Lund and O. Roche-Newton, On distinct perpendicular bisectors and pinned distances i n ﬁnite ﬁelds, Finite Fields Appl. 37(2016), 240-264

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D. Hart, A. Iosevich, D. Koh, and M. Rudnev, Averages over hyperplanes, sum-product theory in vector sp aces over ﬁnite ﬁelds and the Erd˝ os–Falconer distance conjecture, Trans. Amer. Math. Soc. 363, (2011), no. 6, 3255-3275

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Iosevich and M

A. Iosevich and M. Rudnev, Erd˝ os distance problem in vector spaces over ﬁnite ﬁelds , Trans. Am. Math. Soc. 359 (2007), 6127-6142

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Iwaniec, and E

H. Iwaniec, and E. Kowalski, Analytic Number Theory, Colloquium Publications 53 (2004)

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D. Koh, T. Pham, and L.A. Vinh, Extension theorems and a connection to the Erd˝ os-Falconer distance problem over ﬁnite ﬁelds, J. Funct. Anal. 281 (2021), no. 8, 109137, 54 pp

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Koh and C.Y

D. Koh and C.Y. Shen, Sharp extension theorems and Falconer distance problems fo r algebraic curves in two dimensional vector spaces over ﬁnite ﬁelds , Rev. Mat. Iberoam. 28 (2012), no. 1, 157-178

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Lidl and H

R. Lidl and H. Niederreiter, Finite ﬁelds, Cambridge University Press, (1997)

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Murphy, G

B. Murphy, G. Petridis, T. Pham, M. Rudnev, and S. Stevens , On the Pinned Distances Problem in Positive Characteristic, J. Lond. Math. Soc.(2) 105 (2022), no. 1, 469-499

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Shparlinski, On the set of distance between two sets over ﬁnite ﬁelds , International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 59482, Pa ges 1–5

I. Shparlinski, On the set of distance between two sets over ﬁnite ﬁelds , International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 59482, Pa ges 1–5. College of Engineering and Technology, American Universit y of the Middle East, Kuw ait Email ad...

2006

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [12]

Iosevich, D

A. Iosevich, D. Koh, and F. Rakhmonov, The quotient set of the quadratic distance set over ﬁnite ﬁel ds, Forum Math. 36 (2024), no. 5, 1341-1358

work page 2024

[2] [1]

Ahmadi and A

O. Ahmadi and A. Mohammadian, Sets with many pairs of orthogonal vectors over ﬁnite ﬁelds , Finite Fields Appl. 37 (2016), 179-192

work page 2016

[3] [2]

Bennett, D

M. Bennett, D. Hart, A. Iosevich, J. Pakianathan, and M. Ru dnev, Group actions and geometric combinatorics in Fd q, Forum Math. 29(2017), no. 1, 91-110

work page 2017

[4] [3]

Borges, A

T. Borges, A. Iosevich, and Y. Ou, A singular variant of the Falconer distance problem , preprint (2023), arXiv:2306.05247

arXiv 2023

[5] [4]

Bourgain, N

J. Bourgain, N. Katz, and T. Tao, A sum-product estimate in ﬁnite ﬁelds, and applications , Geometric & Func- tional Analysis, 14 (2004), no. 2, 27-57

work page 2004

[6] [5]

Chapman, M

J. Chapman, M. Burak Erdo˘ gan, D. Hart, A. Iosevich, and D. Koh, Pinned distance sets, k-simplices, Wolﬀ ’s exponent in ﬁnite ﬁelds and sum-product estimates , Math Z. 271 (2012), no. 1, 63-93

work page 2012

[7] [6]

Cheong, D

D. Cheong, D. Koh, and T. Pham, An asymmetric bound for sum of distance sets , Tr. Mat. Inst. Steklova 314 (2021), 290-300

work page 2021

[8] [7]

Grove, Classical Groups and Geometric Algebra , Grad

L.C. Grove, Classical Groups and Geometric Algebra , Grad. Stud. Math. vol.39, American Mathematical Society, Providence, RI, 2002. 12

work page 2002