A Sidon set constant
Description of constant
$C_{5}$ is the smallest constant such that Sidon sets in ${1,\dots,N}$ have cardinality $N^{1/2} + (C_{5} + o(1))N^{1/4}$.
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Carter, Georgiev, Gomez–Serrano, Hunter, O’Bryant, Tao, Wagner (unpublished, 2025) |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 |
Known lower bounds
| Bound |
Reference |
Comments |
| $0$ |
[Si38] |
|
- 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).
- [CHO25] Carter, D. and Hunter, Z. and O’Bryant, K., On the diameter of finite {S}idon sets. Acta Math. Hungar. (2025), 108–126.
- [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.
- [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.