A Sidon set constant

Description of constant

$C_{5a}$ is the smallest constant such that Sidon sets in $\{1,\dots,N\}$ have cardinality $N^{1/2} + (C_{5a} + o(1))N^{1/4}$.

Known upper bounds

Bound	Reference	Comments
$1$	[ET41], [Li69]
$0.998$	[BFR21]
$0.99703$	[OBO22]
$0.98183$	[CHO25]
$0.97633$	Carter, Georgiev, Gomez–Serrano, Hunter, O’Bryant, Tao, Wagner (unpublished, 2025)	AlphaEvolve
$0.98173$*	[St26]	Search-and-certification pipeline, improving the then-published $0.98183$ of [CHO25]. Superseded by the rows below, and by the $0.97633$ above, which it predates by three months but does not improve.
$0.9435$	[HZ26]	$\gamma_0 = 0.94349\dots < 0.9435$, by a vector-valued smoothing method combining several discrete smoothing kernels with boundary weights. Stated against the published $0.98183$ of [CHO25]; the unpublished $0.97633$ above is not accounted for there.
$0.94325$*	[Mad26]	$\gamma = 0.943244449828\dots < 0.94325$, from a jointly optimised certificate inside the [HZ26] framework; a fallback certificate at that paper’s own $m=32$, $L=4$ grid gives $0.943260253\dots$. Two independent exact-rational checkers, a mutation-test harness, and a Lean 4 + Mathlib kernel check of all $161$ covering inequalities with no `sorry`. The Lean covers the inequalities, not [HZ26, Lemma 2.1], which the author identifies as the load-bearing unreviewed input. Announced on the forum; not posted to arXiv.
$0.9428095$*	[Akw26]	Symmetric certificates of the [HZ26] framework built from centred uniform kernels, giving $\gamma < 0.942809493$ at $m=1024$, checked in exact arithmetic on $6145$ covering inequalities. The forum comment of 5 October 2026 announced $0.9428096$ from an earlier draft; the figure here is from version 6 of the report, and the file name linked in that comment no longer exists in the repository. Unrefereed and not on arXiv.
$0.94304$	[COT26]	Corollary 1.2: a Sidon set $\mathcal{A}\subseteq[1,N]$ with $k\ge 483$ elements satisfies $k \le N^{1/2} + 0.94304\,N^{1/4} + 0.44466$, equivalently $\mathrm{diam}(\mathcal{A}) \ge k^2 - 1.88607\,k^{3/2}$. The fully explicit bound valid for every $k \ge 1$ is $\lvert\mathcal{A}\rvert \le N^{1/2} + 0.945\,N^{1/4} + 0.447$; since $C_{5a}$ is asymptotic, the large-$k$ constant is the one recorded.

Known lower bounds

Bound	Reference	Comments
$0$	[Si38]

Additional comments and links

On the constant attributed to [CHO25]. [COT26, §1] cites that paper as giving $\mathrm{diam}(\mathcal{A}) \ge k^2 - 1.98183\,k^{3/2} - O(k)$, which would correspond to $C_{5a} \le 0.990915$. The figure recorded above is [CHO25]’s own: its abstract states $b \le 1.96365$ and, equivalently, $n^{1/2} + 0.98183\,n^{1/4} + O(1)$, and $1.96365/2 = 0.981825$. [HZ26] also describes $0.98183$ as the previously published coefficient. The $1.98183$ appears to be a slip in [COT26]; nothing else depends on it.
A barrier for the smoothing method, not a lower bound on $C_{5a}$. [Akw26] argues that every certificate of the [HZ26] vector-valued smoothing framework, symmetric or two-sided, has $\gamma \ge 2\sqrt{2}/3 = 0.9428090415\dots$, and that the symmetric framework attains it; the same value is reported for a pair-level moment relaxation and a three-point relaxation. If correct this caps the method about $5\times 10^{-7}$ below the $0.9428095$ row, and it does not bound $C_{5a}$ from below — a different argument could go under it. The lower bound recorded on this page remains $0$ [Si38]. None of these barrier proofs has been refereed.
This is part of Erdős problem #30.
A survey of the literature can be found at [OBO4].

References

[BFR21] Balogh, J. and F"{u}redi, Z. and Roy, S., An upper bound on the size of Sidon sets. arXiv:2103.15850 (2021).
[Akw26] Akwei, John, A data science analysis of Erdős problem 30 and The limit of vector-valued smoothing for Sidon sets. Reports in the repository johnakwei/Science (version 6, 2026); announced in the Erdős Problems forum thread for problem 30, 5 October 2026.
[CHO25] Carter, D. and Hunter, Z. and O’Bryant, K., On the diameter of finite {S}idon sets. Acta Math. Hungar. (2025), 108–126.
[COT26] Carter, Daniel; O’Bryant, Kevin; Tao, Terence, On the diameter and distribution of Sidon sets. arXiv:2610.36207 (26 October 2026).
[ET41] Erd\H{o}s, P. and Tur'{a}n, P., On a problem of Sidon in additive number theory, and on some related problems. J. London Math. Soc. (1941), 212-215.
[HZ26] Hou, Jianfeng; Zhao, Hongbin, An improved upper bound for finite Sidon sets via vector-valued smoothing. arXiv:2607.01169 (1 July 2026; v3, 4 September 2026).
[Mad26] Madeiros, Josh, improved certificate in the Hou–Zhao framework, with exact-rational and Lean 4 verifiers; announced in the Erdős Problems forum thread for problem 30, 17 August 2026.
[Li69] Lindstr"{o}m, B., An inequality for $B_{2}$-sequences. J. Combinatorial Theory (1969), 211-212.
[OBO4] O’Bryant, Kevin, A complete annotated bibliography of work related to {S}idon sequences. Electron. J. Combin. (2004), 39.
[OBO22] O’Bryant, K., On the size of finite Sidon sets. arXiv:2207.07800 (2022).
[Si38] Singer, James, A theorem in finite projective geometry and some applications to number theory. Trans. Amer. Math. Soc. (1938), 377–385.
[St26] Stoltz, Rynaldo, Erdős–Sidon records, github.com/rynald0cst0ltziam/Erdos-Sidon-Records-30; announced in the Erdős Problems forum thread for problem 30, 17 February 2026.

← Back to the table of constants