pith. sign in

arxiv: 2606.29965 · v1 · pith:C56OK6IHnew · submitted 2026-06-29 · 🧮 math.CO · math.CA

A Delsarte Linear Programming Approach to the ErdH{o}s--Falconer Distance Problem over Finite Fields

Tao Zhang This is my paper

Pith reviewed 2026-06-30 05:36 UTC · model grok-4.3

classification 🧮 math.CO math.CA
keywords Erdős-Falconer distance problemfinite fieldsDelsarte linear programmingquadratic formsassociation schemesGauss sumsKloosterman sumsdistance sets
0
0 comments X

The pith

For point sets larger than C q to the n over 2 plus one third, the quadratic distance set covers a positive fraction of all field elements.

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

The paper applies the Delsarte linear programming method to the Erdős-Falconer distance problem in finite fields of odd characteristic. It proves that for any fixed alpha between 0 and 1/2, when the field size q is large enough, any set E whose cardinality is at least C_alpha times q to the power n over 2 plus one third must have its quadratic distance set Delta_Q(E) contain more than alpha times q distinct values. This shows that the distances occupy a positive proportion of the whole field F_q. The bound holds uniformly for every nondegenerate quadratic form on an even-dimensional space and improves the previous general exponent of n plus one over 2 down to n over 2 plus one third in the Euclidean case for even dimensions at least 4.

Core claim

By analyzing the eigenvalues of the association scheme whose relations are the level sets of a nondegenerate quadratic form Q via Gauss sums and Kloosterman sums, the authors construct a feasible solution to the Delsarte linear program that forces any sufficiently large point set E to realize a positive proportion of all possible quadratic distances.

What carries the argument

The Delsarte linear program on the association scheme whose classes are the level sets of the quadratic form Q, using eigenvalues derived from Gauss sums and Kloosterman sums to obtain a feasible dual solution.

If this is right

  • The quadratic distance set Delta_Q(E) contains a positive proportion of the elements of F_q.
  • The result applies uniformly to every nondegenerate quadratic form on even-dimensional spaces.
  • In the Euclidean case the exponent improves from (n+1)/2 to n/2 + 1/3 for every even n at least 4 over arbitrary odd-characteristic finite fields.
  • The distance set estimate holds once the field size exceeds a constant depending only on alpha.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same linear-programming technique on association schemes might be tried on other finite-field distance or sumset problems that admit natural partitions into level sets.
  • Finding stronger feasible solutions or tighter eigenvalue bounds could push the exponent below n/2 + 1/3.
  • The method links the distance problem directly to character-sum estimates, so new sum bounds would immediately translate into better distance results.

Load-bearing premise

The eigenvalues coming from the Gauss and Kloosterman sums on the quadratic level-set association scheme admit a feasible solution to the Delsarte linear program that achieves the 1/3 exponent improvement.

What would settle it

An explicit construction of a set E with size roughly q to the n over 2 plus one third whose quadratic distances occupy only o(q) values in a large finite field would disprove the bound.

read the original abstract

We introduce a Delsarte linear programming approach to the finite field Erd\H{o}s--Falconer distance problem. Let \(q\) be an odd prime power, let \(n\) be even, and let \(Q\) be a non-degenerate quadratic form on \(\mathbb{F}_q^n\). For \(E\subset \mathbb{F}_q^n\), define \[ \Delta_Q(E)=\{Q(x-y):\ x,y\in E\}. \] We prove that, for every fixed \(0<\alpha<\frac{1}{2}\), there exist constants \(C_\alpha>0\) and \(q_\alpha\) such that if \(q\ge q_\alpha\) and $|E|\ge C_\alpha q^{\frac n2+\frac13},$ then \[ |\Delta_Q(E)|>1+\alpha(q-1). \] In particular, \(\Delta_Q(E)\) contains a positive proportion of the elements of \(\mathbb{F}_q\), and hence \(|\Delta_Q(E)|\gg q\). Our result applies uniformly to all non-degenerate quadratic forms in even-dimensional finite field vector spaces. In the Euclidean case \[ Q(x)=x_1^2+\cdots+x_n^2, \] it improves, for every even \(n\ge 4\) over arbitrary finite fields, the general exponent \(\frac{n+1}{2}\) obtained by Iosevich and Rudnev to $\frac n2+\frac13.$ The proof is based on the association scheme arising from the level sets of \(Q\). By analyzing the corresponding eigenvalues through Gauss sums and Kloosterman sums, we construct a suitable feasible solution to the Delsarte linear program. This provides a new algebraic-combinatorial method for obtaining distance set estimates over finite fields.

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

2 major / 2 minor

Summary. The manuscript introduces a Delsarte linear programming method on the association scheme whose classes are the level sets of a non-degenerate quadratic form Q in even dimension n over F_q. It claims that for any fixed 0<α<1/2 there exist C_α and q_α such that |E|≥C_α q^{n/2+1/3} implies |Δ_Q(E)|>1+α(q−1), improving the prior general exponent (n+1)/2 in the Euclidean case for even n≥4.

Significance. If the claimed feasible solution to the Delsarte LP is correctly exhibited and verified, the result supplies a uniform improvement in the distance-set exponent that holds for every non-degenerate Q. The algebraic-combinatorial approach via Gauss and Kloosterman sums on the scheme eigenvalues is a new technique for this problem and could extend to other association-scheme bounds in finite-field combinatorics.

major comments (2)
  1. [Proof of the main theorem (eigenvalue analysis and LP construction)] The central claim rests on the existence of an explicit feasible point in the Delsarte LP whose objective value yields the exponent n/2+1/3. The abstract states that such a point is constructed from the eigenvalues expressed via Gauss sums (diagonal) and Kloosterman sums (off-diagonal), but neither the explicit test function (or dual vector) nor the verification that all constraints are satisfied for large q appears in the provided text; without this, the improvement over the (n+1)/2 threshold cannot be checked.
  2. [Eigenvalue formulas and LP feasibility verification] The Kloosterman-sum estimates used to certify feasibility must hold uniformly for every non-degenerate Q and for even n. The manuscript should display the precise asymptotic bounds (including error terms and the range of q) that guarantee the chosen point remains feasible and produces a positive proportion of distances; any gap in the sign or magnitude control for even n would collapse the exponent back to the previous bound.
minor comments (2)
  1. Notation for the association scheme classes and the precise definition of the Delsarte LP (primal/dual) should be stated once at the beginning of the proof section for readability.
  2. The statement of the main theorem should explicitly record that the constants C_α and q_α depend on α and n but are independent of the choice of Q.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for identifying areas where the LP construction and eigenvalue analysis require greater explicitness. We will revise the manuscript to address these points directly.

read point-by-point responses

  1. Referee: [Proof of the main theorem (eigenvalue analysis and LP construction)] The central claim rests on the existence of an explicit feasible point in the Delsarte LP whose objective value yields the exponent n/2+1/3. The abstract states that such a point is constructed from the eigenvalues expressed via Gauss sums (diagonal) and Kloosterman sums (off-diagonal), but neither the explicit test function (or dual vector) nor the verification that all constraints are satisfied for large q appears in the provided text; without this, the improvement over the (n+1)/2 threshold cannot be checked.

    Authors: The construction of the feasible dual vector is given in Section 3 via the normalized eigenvalues λ_i expressed through Gauss sums on the diagonal classes and Kloosterman sums on the off-diagonal classes. To improve readability and verifiability, the revised version will include an explicit formula for the test function f together with a dedicated verification subsection showing that all Delsarte constraints hold with the claimed objective value for q sufficiently large, thereby confirming the exponent n/2 + 1/3. revision: yes

  2. Referee: [Eigenvalue formulas and LP feasibility verification] The Kloosterman-sum estimates used to certify feasibility must hold uniformly for every non-degenerate Q and for even n. The manuscript should display the precise asymptotic bounds (including error terms and the range of q) that guarantee the chosen point remains feasible and produces a positive proportion of distances; any gap in the sign or magnitude control for even n would collapse the exponent back to the previous bound.

    Authors: The eigenvalue formulas already incorporate the standard Weil bounds |K(χ,ψ)| ≤ 2q^{1/2} for Kloosterman sums, which are uniform over all non-degenerate quadratic forms because the association scheme is isomorphic for any such Q. The revised manuscript will add an appendix stating the precise error terms O(q^{-1/2}) and the explicit threshold q ≥ q_α (depending only on α and n) under which the chosen point satisfies all inequalities strictly, ensuring |Δ_Q(E)| > 1 + α(q-1) for |E| ≥ C_α q^{n/2 + 1/3}. revision: yes

Circularity Check

0 steps flagged

No circularity: feasible Delsarte LP solution constructed from external Gauss/Kloosterman eigenvalue estimates

full rationale

The derivation computes the association-scheme eigenvalues via standard Gauss and Kloosterman sum formulas (external to the target bound), then exhibits an explicit feasible vector for the Delsarte LP whose objective yields the |E| ≳ q^{n/2 + 1/3} threshold. This construction is self-contained against the external sum estimates and does not reduce the final statement to a fitted parameter, self-definition, or load-bearing self-citation chain. The method is uniform over non-degenerate Q and improves the prior (n+1)/2 exponent without renaming known results or smuggling ansatzes.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The argument rests on standard facts about association schemes and character sums over finite fields; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • standard math The level sets of a non-degenerate quadratic form Q define an association scheme on F_q^n whose eigenvalues can be expressed via Gauss and Kloosterman sums.
    Invoked to set up the Delsarte linear program (abstract, proof paragraph).
  • domain assumption The Delsarte linear programming bound applies directly to the distance-set problem once the scheme eigenvalues are known.
    Central modeling step that converts the combinatorial question into an LP feasibility problem.

pith-pipeline@v0.9.1-grok · 5878 in / 1547 out tokens · 38973 ms · 2026-06-30T05:36:55.266181+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

22 extracted references · 4 canonical work pages · 3 internal anchors

  1. [1]

    Bennett, D

    M. Bennett, D. Hart, A. Iosevich, J. Pakianathan, and M. Rudnev. Group actions and geometric combinatorics inF d q.Forum Math., 29(1):91–110, 2017

    2017

  2. [2]

    Bourgain, N

    J. Bourgain, N. Katz, and T. Tao. A sum-product estimate in finite fields, and applications. Geom. Funct. Anal., 14(1):27–57, 2004. 14

    2004

  3. [3]

    A. E. Brouwer, A. M. Cohen, and A. Neumaier.Distance-regular graphs, volume 18 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1989

    1989

  4. [4]

    Chapman, M

    J. Chapman, M. B. 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(1- 2):63–93, 2012

    2012

  5. [5]

    Additive structures imply more distances in $\mathbb{F}_q^d$

    D. Cheong, G. Ge, D. Koh, T. Pham, D. Tran, and T. Zhang. Additive structures imply more distances inF d q. arXiv: 2510.26364

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    Delsarte

    P. Delsarte. An algebraic approach to the association schemes of coding theory.Philips Res. Rep. Suppl., (10):vi+97, 1973

    1973

  7. [7]

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

    work page arXiv

  8. [8]

    J. M. Fraser.L p averages of the Fourier transform in finite fields. arXiv: 2407.08589

    work page internal anchor Pith review Pith/arXiv arXiv

  9. [9]

    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

    2020

  10. [10]

    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.Trans. Amer. Math. Soc., 363(6):3255–3275, 2011

    2011

  11. [11]

    Iosevich and M

    A. Iosevich and M. Rudnev. Erd˝ os distance problem in vector spaces over finite fields. Trans. Amer. Math. Soc., 359(12):6127–6142, 2007

    2007

  12. [12]

    D. Koh, M. Q. Pham, and T. Pham. Structural theorems on the distance sets over finite fields.Forum Math., 35(4):925–938, 2023

    2023

  13. [13]

    D. Koh, T. Pham, C.-Y. Shen, and L. A. Vinh. A sharp exponent on sum of distance sets over finite fields.Math. Z., 297(3-4):1749–1765, 2021

    2021

  14. [14]

    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(8):Paper No. 109137, 54, 2021

    2021

  15. [15]

    Koh and H.-S

    D. Koh and H.-S. Sun. Distance sets of two subsets of vector spaces over finite fields.Proc. Amer. Math. Soc., 143(4):1679–1692, 2015

    2015

  16. [16]

    Mockenhaupt and T

    G. Mockenhaupt and T. Tao. Restriction and Kakeya phenomena for finite fields.Duke Math. J., 121(1):35–74, 2004

    2004

  17. [17]

    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(1):469–499, 2022

    2022

  18. [18]

    Pham and A

    T. Pham and A. Suk. On the structure of distance sets over prime fields.Proc. Amer. Math. Soc., 148(8):3209–3215, 2020. 15

    2020

  19. [19]

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

    2019

  20. [20]

    On the prime field spherical restriction conjecture in four dimensions: breaking the Stein-Tomas exponent and applications

    T. Pham and B. Xue. On the prime field spherical restriction conjecture in four dimensions: breaking the Stein-Tomas exponent and applications. arXiv: 2606.03627

    work page internal anchor Pith review Pith/arXiv arXiv

  21. [21]

    J. v. Neumann. Zur Theorie der Gesellschaftsspiele.Math. Ann., 100(1):295–320, 1928

    1928

  22. [22]

    T. Wolff. Decay of circular means of Fourier transforms of measures.Internat. Math. Res. Notices, (10):547–567, 1999. 16

    1999