An autocorrelation constant related to Sidon sets

Description of constant

$C_{1a}$ is the largest constant for which one has $$\underset{-1/2\le t\le 1/2}{\max }{\int }_{\mathbb{R}}f(t-x)f(x)dx\ge C_{1a}{\left({\int }_{-1/4}^{1/4}f(x)dx\right)}^2$$ for all non-negative $f:\mathbb{R}\to \mathbb{R}$.

Known upper bounds

Known lower bounds

Additional comments and links

• Damek Davis's meta-analysis of this problem.
• AlphaEvolve repository page for this problem. This repository also contains pages for some similar autocorrelation constants, see here, here, and here. See also the page here for the minimum overlap problem.

References

• [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 $C_{1a}$, 2026. https://github.com/AndreiPiterbarg/sidon-autocorrelation
• [SS2002] Schinzel, A.; Schmidt, W. M.. Comparison of $L^1$ and $L^{\infty }$ 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

What counts as progress

• A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
• A formal proof (Lean) of a known bound, or a precise error in a claimed one.
• New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.