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OP-12001OpenNumber theory

Sidon set size constant (C5aC_{5a})

Sidon set size constant: determine C5aC_{5a} or improve its bounds; the repository records 0 (lower) and 0.97633 (upper).

MSC 11B83 · Special sequences and polynomials

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

C5aC_{5a} is the smallest constant such that Sidon sets in {1,…,N}\{1,\dots,N\} have cardinality N1/2+(C5a+o(1))N1/4N^{1/2} + (C_{5a} + o(1))N^{1/4}.

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Background

References (from the repository page)

  • [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 B2B_{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.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0; best known upper bound 0.97633. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
11 [ET41], [Li69]
0.9980.998 [BFR21]
0.997030.99703 [OBO22]
0.981830.98183 [CHO25]
0.976330.97633 Carter, Georgiev, Gomez--Serrano, Hunter, O'Bryant, Tao, Wagner (unpublished, 2025) AlphaEvolve

Known lower bounds

Bound Reference Comments
00 [Si38]
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Sidon set size constant ($C_{5a}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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