Vector-valued smoothing for finite Sidon sets

Hongbin Zhao; Jianfeng Hou

arxiv: 2607.01169 · v1 · pith:PAXB76MMnew · submitted 2026-07-01 · 🧮 math.CO

Vector-valued smoothing for finite Sidon sets

Jianfeng Hou , Hongbin Zhao This is my paper

Pith reviewed 2026-07-02 09:57 UTC · model grok-4.3

classification 🧮 math.CO

keywords Sidon setsupper boundssmoothing kernelsconvolution inequalitiesadditive combinatoricsfinite setsvector-valued inequalitiesboundary majorants

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The pith

The largest Sidon subset of {0, ..., N-1} has size at most sqrt(N) + 0.94601 N^{1/4} + O(1).

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

The paper proves a new upper bound on F(N), the maximum cardinality of a Sidon subset inside the initial segment of length N. A Sidon set requires all pairwise sums to be distinct. The bound improves the coefficient on the N^{1/4} term from 0.97633 to 0.94601 and lies numerically below a previously tentative value near 0.947. The proof distributes the construction of a boundary majorant across several smoothing kernels whose L2 energies are then averaged inside a vector-valued convolution inequality. The final constant arises from a finite rational certificate that a short program verifies using exact arithmetic.

Core claim

We prove F(N) ≤ N^{1/2} + 0.94601 N^{1/4} + O(1). The argument rests on a vector-valued convolution inequality in which multiple smoothing kernels together produce a boundary majorant while their L2 energies are averaged. The analytic reduction that converts this inequality into the stated bound on F(N) is elementary. The numerical constant is supplied by a finite rational certificate checked by a short program that uses only exact arithmetic.

What carries the argument

Vector-valued convolution inequality in which several smoothing kernels share the task of producing a boundary majorant and their L2 energies are averaged.

If this is right

The coefficient multiplying N^{1/4} drops from the prior value 0.97633 to 0.94601.
The new upper bound lies numerically below the previously reported tentative value near 0.947.
The vector-valued averaging step allows the majorant task to be shared across kernels without altering the elementary reduction to a bound on F(N).
The constant is obtained from a finite rational certificate verified by exact arithmetic alone.

Where Pith is reading between the lines

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

The same vector-valued averaging technique could be applied to other upper-bound problems that currently rely on single-kernel majorants.
Increasing the number of kernels in the family might produce still smaller coefficients while preserving the elementary reduction.
The method isolates the search for good constants to a finite linear-programming-style certificate that can be checked independently of the analytic steps.

Load-bearing premise

The chosen family of smoothing kernels produces a valid boundary majorant whose averaged L2 energies yield the stated coefficient after the elementary analytic reduction.

What would settle it

An explicit Sidon subset of {0,1,...,N-1} whose size exceeds N^{1/2} + 0.94601 N^{1/4} + C for sufficiently large N and some fixed C, or a computation showing that the rational certificate fails to satisfy the required inequality.

read the original abstract

Let $F(N)$ denote the largest cardinality of a Sidon subset of $\{0, 1, \dots, N - 1\}$. We prove \[ F(N) \le N^{1/2} + 0.94601 N^{1/4} + O(1). \] This improves the recently announced coefficient $0.97633$ obtained by Carter, Georgiev, G\'{o}mez-Serrano, Hunter, O'Bryant, Tao and Wagner. It is also very close to, and numerically below, the tentatively reported value of approximately $0.947$. The argument is based on a vector-valued convolution inequality: several smoothing kernels share the task of producing a boundary majorant, while their $L^2$ energies are averaged. The analytic reduction is elementary. The final constant is supplied by a finite rational certificate, verified by a short program using exact arithmetic only.

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.

Desk Editor's Note private letter to a colleague

They improve the explicit Sidon constant to 0.94601 via vector-valued kernels and an exact-arithmetic certificate.

read the letter

The main takeaway is that this paper lowers the explicit coefficient in the Sidon bound from 0.97633 to 0.94601. The new bound is F(N) ≤ N^{1/2} + 0.94601 N^{1/4} + O(1), and it sits just below the tentative 0.947 value mentioned in the abstract.

They achieve this by moving to a vector-valued convolution inequality where several smoothing kernels work together to produce the boundary majorant. Their L2 energies are then averaged. The analytic reduction stays elementary, and the final number comes from a finite rational certificate checked with exact arithmetic in a short program.

That combination is what is actually new. The vector-valued step lets them extract a tighter constant without altering the core smoothing strategy, and the certificate avoids any fitting or self-referential steps.

The soft spots are limited. The abstract gives the outline but not the explicit kernels or the program itself, so a reader would want to see the code to confirm the arithmetic. The stress-test note finds no internal inconsistency or unverifiable assumption, which matches what the abstract describes. If the certificate is as claimed, the bound holds.

This is for people who track explicit constants in additive combinatorics or related problems in harmonic analysis. A reader already following the Sidon question or similar smoothing arguments would pick up the method and the improved number.

It deserves a serious referee. The claim is concrete, the method is elementary on its face, and the verification route is reproducible. A referee can check the certificate directly.

Referee Report

1 major / 0 minor

Summary. The paper proves that the largest Sidon subset F(N) of {0,1,...,N-1} satisfies F(N) ≤ N^{1/2} + 0.94601 N^{1/4} + O(1). The argument proceeds by an elementary analytic reduction to a vector-valued convolution inequality in which several smoothing kernels jointly produce a boundary majorant; the coefficient 0.94601 is obtained by averaging the L² energies of these kernels and is supplied by a finite rational certificate verified by exact-arithmetic computation. This improves the coefficient 0.97633 from Carter et al.

Significance. If the claimed inequality holds, the result tightens the best known upper bound on finite Sidon sets and brings the secondary-term coefficient numerically below the tentatively reported value ≈0.947. The vector-valued smoothing technique, together with the use of an exact-arithmetic certificate, supplies a reproducible and machine-checkable constant; both features strengthen the contribution and may extend to other problems in additive combinatorics.

major comments (1)

[Abstract (and any section describing the certificate)] The numerical constant 0.94601 is the load-bearing output of the finite rational certificate. The manuscript must exhibit either the explicit certificate (as a list of rationals) or the short program together with its exact-arithmetic output so that the coefficient can be independently recomputed; without this material the central claim cannot be verified from the text alone.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation of minor revision. We address the single major comment below.

read point-by-point responses

Referee: [Abstract (and any section describing the certificate)] The numerical constant 0.94601 is the load-bearing output of the finite rational certificate. The manuscript must exhibit either the explicit certificate (as a list of rationals) or the short program together with its exact-arithmetic output so that the coefficient can be independently recomputed; without this material the central claim cannot be verified from the text alone.

Authors: We agree that the manuscript as currently written mentions the certificate and program but does not exhibit them. In the revised manuscript we will add an appendix containing the short program (in a language supporting exact rational arithmetic) together with its exact-arithmetic output, which consists of the explicit list of rationals that certify the coefficient 0.94601. This addition will make the central claim independently verifiable from the text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via independent certificate

full rationale

The central claim reduces to an elementary analytic reduction from a vector-valued convolution inequality whose L2 energies are averaged across kernels to produce the boundary majorant. The coefficient 0.94601 is obtained from a separate finite rational certificate verified by exact-arithmetic computation, not by fitting to the Sidon bound or by any self-referential step. No self-citations, ansatzes, or uniqueness theorems from the authors appear in the load-bearing chain; the prior coefficient 0.97633 is from unrelated authors. This is the normal case of a self-contained paper whose numerical constant is externally verifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated beyond standard analytic inequalities and a computational certificate.

pith-pipeline@v0.9.1-grok · 5682 in / 1053 out tokens · 26060 ms · 2026-07-02T09:57:09.743029+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 6 canonical work pages

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J. Balogh, Z. Füredi and S. Roy,An upper bound on the size of Sidon sets, Amer. Math. Monthly130(2023), no. 5, 437–445. doi:10.1080/00029890.2023.2176667

work page doi:10.1080/00029890.2023.2176667 2023
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Carter, Z

D. Carter, Z. Hunter and K. O’Bryant,On the diameter of finite Sidon sets, Acta Math. Hungar.175(2025), no. 1, 108–126. doi:10.1007/s10474-024-01499-8

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T. Tao, Comment in the discussion thread for Erdős Problem #30,Erdős Problems, February 17, 2026.https://www.erdosproblems.com/forum/thread/30 Accessed July 1, 2026. Center for Discrete Mathematics, Fuzhou University, Fujian 350108, China Email address:jfhou@fzu.edu.cn Center for Discrete Mathematics, Fuzhou University, Fujian 350108, China Email address:...

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Balogh, Z

J. Balogh, Z. Füredi and S. Roy,An upper bound on the size of Sidon sets, Amer. Math. Monthly130(2023), no. 5, 437–445. doi:10.1080/00029890.2023.2176667

work page doi:10.1080/00029890.2023.2176667 2023

[2] [2]

Carter, Z

D. Carter, Z. Hunter and K. O’Bryant,On the diameter of finite Sidon sets, Acta Math. Hungar.175(2025), no. 1, 108–126. doi:10.1007/s10474-024-01499-8

work page doi:10.1007/s10474-024-01499-8 2025

[3] [3]

Sidon set size constant,

D. Davis, P. Ivanisvili, T. Tao and contributors,Optimization Constants in Mathematics, entry C5a, “Sidon set size constant,” online repository, 2026.https://teorth.github. io/optimizationproblems/Accessed July 1, 2026

2026

[4] [4]

Erdős and P

P. Erdős and P. Turán,On a problem of Sidon in additive number theory, and on some related problems, J. London Math. Soc.16(1941), no. 4, 212–215. doi:10.1112/jlms/s1- 16.4.212. VECTOR-V ALUED SMOOTHING FOR SIDON SETS 9

work page doi:10.1112/jlms/s1- 1941

[5] [5]

Lindström,An inequality for B2-sequences, J

B. Lindström,An inequality for B2-sequences, J. Combin. Theory6(1969), no. 2, 211–212. doi:10.1016/S0021-9800(69)80124-9

work page doi:10.1016/s0021-9800(69)80124-9 1969

[6] [6]

O’Bryant,On the size of finite Sidon sets, Ukrainian Math

K. O’Bryant,On the size of finite Sidon sets, Ukrainian Math. J.76(2025), 1352–1368. doi:10.1007/s11253-024-02392-x

work page doi:10.1007/s11253-024-02392-x 2025

[7] [7]

Singer,A theorem in finite projective geometry and some applications to number theory, Trans

J. Singer,A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. Soc.43(1938), no. 3, 377–385. doi:10.1090/S0002-9947- 1938-1501951-4

work page doi:10.1090/s0002-9947- 1938

[8] [8]

Tao, Comment in the discussion thread for Erdős Problem #30,Erdős Problems, February 17, 2026.https://www.erdosproblems.com/forum/thread/30 Accessed July 1, 2026

T. Tao, Comment in the discussion thread for Erdős Problem #30,Erdős Problems, February 17, 2026.https://www.erdosproblems.com/forum/thread/30 Accessed July 1, 2026. Center for Discrete Mathematics, Fuzhou University, Fujian 350108, China Email address:jfhou@fzu.edu.cn Center for Discrete Mathematics, Fuzhou University, Fujian 350108, China Email address:...

2026