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Sidon set autocorrelation constant (C1aC_{1a})

Sidon set autocorrelation constant: determine C1aC_{1a} or improve its bounds; the repository records 1.2802 (1.292*) (lower) and 1.502862 (upper).

MSC 42A85 · Convolution, factorization for one variable harmonic analysisMSC 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

C1aC_{1a} is the largest constant for which one has

max⁡−1/2≤t≤1/2∫Rf(t−x)f(x) dx≥C1a(∫−1/41/4f(x) dx)2\max_{-1/2 \leq t \leq 1/2} \int_{\mathbb{R}} f(t-x) f(x)\ dx \geq C_{1a} \left(\int_{-1/4}^{1/4} f(x)\ dx\right)^2

for all non-negative f ⁣:R→Rf \colon \mathbb{R} \to \mathbb{R}.

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Background

References (from the repository page)

  • [GGSWT2025] Georgiev, Bogdan; Gómez-Serrano, Javier; Tao, Terence; Wagner, Adam Zsolt. Mathematical exploration and discovery at scale. arXiv:2511.02864
  • [CS2017] Cloninger, Alexander; Steinerberger, Stefan. On suprema of autoconvolutions with an application to Sidon sets. Proc. Amer. Math. Soc. 145, No. 8, 3191–3200 (2017). arXiv:1403.7988
  • [MO2004] Martin, Greg; O’Bryant, Kevin. The symmetric subset problem in continuous Ramsey theory. Exp. Math. 16, No. 2, 145-165 (2007). arXiv:math/0410004
  • [MO2009] Martin, Greg; O’Bryant, Kevin. The supremum of autoconvolutions, with applications to additive number theory. Ill. J. Math. 53, No. 1, 219-235 (2009). arXiv:0807.5121
  • [MV2009] Matolcsi, Máté; Vinuesa, Carlos. Improved bounds on the supremum of autoconvolutions. J. Math. Anal. Appl. 372, No. 2, 439-447 (2010). arXiv:0907.1379
  • [PBV2026] Piterbarg, Andrei; Bajaj, Jai; Vincent, Derrick. A multi-scale arcsine lower bound for the Sidon autocorrelation constant C1aC_{1a}, 2026. https://github.com/AndreiPiterbarg/sidon-autocorrelation
  • [SS2002] Schinzel, A.; Schmidt, W. M.. Comparison of L1L^1 and L∞L^\infty norms of squares of polynomials. Acta Arith. 104, No. 3, 283-296 (2002).
  • [WSZXRYHHMPCHCWDS2025] Wang, Yiping; Su, Shao-Rong; Zeng, Zhiyuan; Xu, Eva; Ren, Liliang; Yang, Xinyu; Huang, Zeyi; He, Pengcheng; Cheng, Hao; Chen, Weizhu; Wang, Shuohang; Du, Simon Shaolei; Shen, Yelong. ThetaEvolve: Test-time Learning on Open Problems. arXiv:2511.23473
  • [IX2026] Ivanisvili, Paata; Xie, Xinyuan. Grokability in five inequalities. 2026. arXiv:2605.05193
  • [YKLBMWKCZGS2026] Yuksekgonul, Mert; Koceja, Daniel; Li, Xinhao; Bianchi, Federico; McCaleb, Jed; Wang, Xiaolong; Kautz, Jan; Choi, Yejin; Zou, James; Guestrin, Carlos; Sun, Yu. Learning to Discover at Test Time, 2026.
  • [T2026] Together AI. Einsteinarena-new-sota: State-of-the-art results on open math problems, 2026. URL https://github.com/togethercomputer/EinsteinArena-new-SOTA.
  • [YLTLYSTYLLGDHZSWZSHMELCZX2026] Haotian Ye, Haowei Lin, Jingyi Tang, Yizhen Luo, Caiyin Yang, Chang Su, Rahul Thapa, Rui Yang, Ruihua Liu, Zeyu Li, Chong Gao, Dachao Ding, Guangrong He, Miaolei Zhang, Lina Sun, Wenyang Wang, Yuchen Zhong, Zhuohao Shen, Di He, Jianzhu Ma, Stefano Ermon, Tongyang Li, Xiaowen Chu, James Zou, Yuzhi Xu, Evaluation-driven Scaling for Scientific Discovery, https://arxiv.org/abs/2604.19341

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1.2802 (1.292*); best known upper bound 1.502862. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
π/2=1.5707963…\pi/2 = 1.5707963\dots [SS2002]
1.509921.50992 [MV2009]
1.50531.5053 [GGSWT2025] May 2025 announcement, AlphaEvolve
1.5031641.503164 [GGSWT2025] Dec 2025 preprint release, AlphaEvolve
1.5031331.503133 [WSZXRYHHMPCHCWDS2025] ThetaEvolve
1.5038711.503871 [YLTLYSTYLLGDHZSWZSHMELCZX2026] SimpleTES
1.5028701.502870 [YKLBMWKCZGS2026] TTT-Discover
1.5028621.502862 [T2026] TogetherAI

Known lower bounds

Bound Reference Comments
11 Trivial
1.1827781.182778 [MO2004]
1.2621.262 [MO2009]
1.27481.2748 [MV2009]
1.281.28 [CS2017]
1.28021.2802 [IX2026]
1.292∗1.292* [PBV2026] Unverified (see README asterisk convention)
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 autocorrelation constant ($C_{1a}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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