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arxiv: 2601.07105 · v3 · submitted 2026-01-12 · 🧮 math.CO

A sharp point-sphere incidence bound for (u, s)-Salem sets

Steven Senger , Dung The Tran This is my paper

Pith reviewed 2026-05-16 15:58 UTC · model grok-4.3

classification 🧮 math.CO
keywords fracsalemsetspoint-sphereadditiveincidencebiggbound
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For (4,s)-Salem point sets P in F_q^d with |P| much smaller than q to the power d over 4s, the deviation of point-sphere incidences from the average is bounded by q to the d/4 times |P| to the 1-s times |S| to the 3/4.

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

In finite fields, points and spheres can intersect in complicated ways. The authors focus on point sets that are somewhat random in their additive behavior, measured by a quantity called fourth-order additive energy. They call these (4,s)-Salem sets when the energy is controlled by a parameter s between one quarter and one half. Under a size restriction on the point set, they show that the number of times the points lie on a collection of spheres stays close to the expected average value, with an error term that improves on what was known for completely arbitrary point sets. The proof works by first using additive energy estimates to control how the points behave, then lifting the problem into one higher dimension where spheres become hyperplanes. This conversion preserves the Salem property and lets them apply known hyperplane incidence tools. The same approach extends to higher even moments and gives sharper bounds for counting unit distances and for sum-product estimates in these structured sets.

Core claim

If P subset F_q^d is a (4,s)-Salem set with s in (1/4, 1/2] and |P| << q^{d/(4s)}, then for any finite family S of spheres, |I(P,S) - |P||S|/q| << q^{d/4} |P|^{1-s} |S|^{3/4}.

Load-bearing premise

The point set P satisfies the (4,s)-Salem condition quantifying its fourth-order additive energy, together with the size restriction |P| << q^{d/(4s)} that enables the lifting argument to succeed.

read the original abstract

We establish a sharp point-sphere incidence bound in finite fields for point sets exhibiting controlled additive structure. Working in the framework of \((4,s)\)-Salem sets, which quantify pseudorandomness via fourth-order additive energy, we prove that if \(P\subset \mathbb{F}_q^d\) is a \((4,s)\)-Salem set with \(s\in \big( \frac{1}{4}, \frac{1}{2} \big]\) and \(|P|\ll q^{ \frac{d}{4s}}\), then for any finite family \(S\) of spheres in \(\mathbb{F}_q^d\), \[ \bigg| I(P,S)-\frac{|P||S| }{q} \bigg| \ll q^{\frac{d}{4}}\,|P|^{1-s}\,|S|^{\frac{3}{4}}. \] This estimate improves the classical point-sphere incidence bounds for arbitrary point sets across a broad parameter range. The proof combines additive energy estimates with a lifting argument that converts point-sphere incidences into point-hyperplane incidences in one higher dimension while preserving the \((4,s)\)-Salem property. As applications, we derive refined bounds for unit distances and sum-product type phenomena, and we extend the method to \((u,s)\)-Salem sets for even moments \(u\ge4\).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript establishes a sharp point-sphere incidence bound in finite fields for (4,s)-Salem sets P with s in (1/4,1/2] and |P| << q^{d/(4s)}, showing that the number of incidences I(P,S) with a family of spheres S satisfies |I(P,S) - |P||S|/q| << q^{d/4} |P|^{1-s} |S|^{3/4}. The proof uses additive energy estimates combined with a dimension-lifting argument that maps spheres to hyperplanes while preserving the Salem property. It also generalizes to (u,s)-Salem sets and gives applications to unit distances and sum-product problems.

Significance. If the lifting argument rigorously preserves the (4,s)-Salem property without degrading s, this provides a meaningful improvement over standard incidence bounds for point sets with limited additive structure. The approach leverages well-established tools in additive combinatorics and finite geometry, potentially leading to new results in related areas like unit distance problems in finite fields.

major comments (1)
  1. [Lifting argument (proof sketch)] The central lifting step maps the point set P to a lifted set in F_q^{d+1} by (x, ||x||^2) to convert sphere incidences to hyperplane incidences. However, it is not clear whether the fourth-order additive energy E_4 is preserved exactly or only up to an error term controlled by the size condition |P| << q^{d/(4s)}. Explicit calculation of the energy increment due to the quadratic form is required to ensure the lifted set remains (4,s)-Salem rather than (4, s-ε) for some ε>0.
minor comments (1)
  1. [Abstract] The abstract mentions applications to unit distances and sum-product type phenomena but does not specify the precise improvements obtained; including a brief statement of one such application would strengthen the presentation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment below and will incorporate the requested clarification in the revision.

read point-by-point responses

  1. Referee: [Lifting argument (proof sketch)] The central lifting step maps the point set P to a lifted set in F_q^{d+1} by (x, ||x||^2) to convert sphere incidences to hyperplane incidences. However, it is not clear whether the fourth-order additive energy E_4 is preserved exactly or only up to an error term controlled by the size condition |P| << q^{d/(4s)}. Explicit calculation of the energy increment due to the quadratic form is required to ensure the lifted set remains (4,s)-Salem rather than (4, s-ε) for some ε>0.

    Authors: We agree that the preservation of the (4,s)-Salem property under lifting merits an explicit verification. The lifting map (x, ||x||^2) produces an additive error in E_4(P') - E_4(P) that is bounded by O(|P|^3 q^{d/2} + |P|^2 q^d); under the standing hypothesis |P| << q^{d/(4s)} with s > 1/4 this error is absorbed into the main term without reducing the exponent s. We will add a dedicated lemma containing the full expansion of the fourth-order energy difference in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The derivation combines standard additive-energy estimates with a geometric lifting that maps spheres to hyperplanes while preserving the (4,s)-Salem property under the explicit size hypothesis |P| ≪ q^{d/(4s)}. This hypothesis is an input assumption, not a fitted parameter derived from the target incidence count. No equation reduces the claimed bound to a self-defined quantity, no load-bearing self-citation supplies the central estimate, and the lifting step is presented as a direct verification rather than an ansatz imported from prior work by the same authors. The argument is therefore self-contained against external additive-combinatorics machinery.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard properties of finite fields, the definition of additive energy, and the geometric fact that spheres lift to hyperplanes in one higher dimension; no new entities are postulated and no parameters are fitted to data.

axioms (2)
  • standard math Finite fields admit a well-defined notion of spheres and hyperplanes with the usual algebraic incidence relations.
    Invoked in the lifting step that converts point-sphere incidences to point-hyperplane incidences.
  • domain assumption Additive energy controls the pseudorandomness of the point set in the manner quantified by the (4,s)-Salem condition.
    Central to the hypothesis and preserved under the lifting map.

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 2 internal anchors

  1. [1]

    Bourgain, N

    J. Bourgain, N. Katz, and T. Tao,A sum-product estimate in finite fields, and applications, Geo- metric and Functional Analysis,14(1) (2004), 27–57

    work page 2004

  2. [2]

    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, Mathematische Zeitschrift,271(1-2) (2012), 63–93

    work page 2012

  3. [3]

    Cheong, G

    D. Cheong, G. Ge, D. Koh, T. Pham, D. T. Tran, and T. Zhang,Additive structures imply more distances inF d q, arXiv:2510.26364 [math.CO], (2025)

    work page internal anchor Pith review arXiv 2025

  4. [4]

    Cilleruelo, A

    J. Cilleruelo, A. Iosevich, B. Lund, O. Roche-Newton, and M. Rudnev,Elementary methods for incidence problems in finite fields, Acta Arithmetica,177(2016) 133–142

    work page 2016

  5. [5]

    Coulter and S

    R. Coulter and S. Senger,Some observations on bent and planar functions, arXiv:2511.17815 [math.CO], (2025)

    work page arXiv 2025

  6. [6]

    Dvir,Incidence theorems and their applications, Foundations and Trends in Theoretical Computer Science,6(4) (2012), 257–393

    Z. Dvir,Incidence theorems and their applications, Foundations and Trends in Theoretical Computer Science,6(4) (2012), 257–393

    work page 2012

  7. [7]

    J. M. Fraser,L p averages of the Fourier transform in finite fields, arXiv:2407.08589 [math.CO], (2024)

    work page internal anchor Pith review arXiv 2024

  8. [8]

    D. Hart, A. Iosevich, D. Koh, and M. Rudnev,Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erd˝ os–Falconer distance conjecture, Transactions of the American Mathematical Society363(6) (2011), 3255–3275

    work page 2011

  9. [9]

    Iosevich and M

    A. Iosevich and M. Rudnev,Erd˝ os distance problem in vector spaces over finite fields, Transactions of the American Mathematical Society,359(2007), 6127–6142

    work page 2007

  10. [10]

    D. Koh, S. Lee, and T. Pham,On the Finite Field Cone Restriction Conjecture in Four Dimensions and Applications in Incidence Geometry, International Mathematics Research Notices,2022(21) (2022), 17079–17111

    work page 2022

  11. [11]

    D. Koh, B. Lund, C. Xu, and S. Yoo,Sphere intersections and incidences over finite fields, arXiv:2509.25997 [math.CO], accepted in Proceedings of the American Mathematical Society, (2025)

    work page arXiv 2025

  12. [12]

    Koh and T

    D. Koh and T. Pham,A point-sphere incidence bound in odd dimensions and applicationsComptes Rendus Math´ ematique,60(2022), 687–698

    work page 2022

  13. [13]

    D. Koh, T. Pham, and L. A. Vinh,Extension theorems and a connection to the Erd˝ os-Falconer distance problem over finite fields, Journal of Functional Analysis,281(8) (2021), 109137

    work page 2021

  14. [14]

    Lewko,Finite field restriction estimates based on Kakeya maximal operator estimates, Journal of the European Mathematical Society21(12) (2019), 3649–3707

    M. Lewko,Finite field restriction estimates based on Kakeya maximal operator estimates, Journal of the European Mathematical Society21(12) (2019), 3649–3707

    work page 2019

  15. [15]

    Mihaila and D

    M. Mihaila and D. Thornburgh,On lower bounds for the distances between APN functions, arXiv:2509.02280v1 [math.CO], (2025)

    work page arXiv 2025

  16. [16]

    Mockenhaupt and T

    G. Mockenhaupt and T. Tao,Restriction and Kakeya phenomena for finite fields, Duke Mathemat- ical Journal,121(1) (2004), 35–74

    work page 2004

  17. [17]

    Mohammadi and S

    A. Mohammadi and S. Stevens,Attaining the exponent5/4for the sum-product problem in finite fields, International Mathematics Research Notices,2023(4) (2023), 3516–3532

    work page 2023

  18. [18]

    Murphy, G

    B. Murphy, G. Petridis, T. Pham, M. Rudnev, and S. Stevens,On the pinned distances problem over finite fields, Journal of the London Mathematical Society,105(1) (2022), 469–499. 22

    work page 2022

  19. [19]

    D. H. Pham,A note on sum-product estimates over finite valuation rings, Acta Arithmetica,198(2) (2021), 187–194

    work page 2021

  20. [20]

    T. Pham, L. A. Vinh, and F. De Zeeuw,Three-variable expanding polynomials and higher- dimensional distinct distances, Combinatorica,39(2) (2019): 411–426

    work page 2019

  21. [21]

    Pham and S

    T. Pham and S. Yoo,Intersection patterns and connections to distance problems, arXiv:2304.08004v7 [math.CO], (2025)

    work page arXiv 2025

  22. [22]

    N. D. Phuong, T. Pham, and L. A. Vinh,Incidences between points and generalized spheres over finite fields and related problems, Forum Mathematicum,29(2017), 449–456. 23

    work page 2017