Description of constant
$C_{1a}$ is the largest constant for which one has
\(\max_{-1/2 \leq t \leq 1/2} \int_{\mathbb{R}} f(t-x) f(x)\ dx \geq C_{1a} \left(\int_{-1/4}^{1/4} f(x)\ dx\right)^2\)
for all non-negative $f \colon \mathbb{R} \to \mathbb{R}$.
Known upper bounds
| Bound |
Reference |
Comments |
| $\pi/2 = 1.57059$ |
[SS2002] |
|
| $1.50992$ |
[MV2009] |
|
| $1.5053$ |
[GGSWT2025] |
May 2025 announcement, AlphaEvolve |
| $1.503164$ |
[GGSWT2025] |
Dec 2025 preprint release, AlphaEvolve |
| $1.503133$ |
[WSZXRYHHMPCHCWDS2025] |
ThetaEvolve |
| $1.503871$ |
[YLTLYSTYLLGDHZSWZSHMELCZX2026] |
SimpleTES |
| $1.502870$ |
[YKLBMWKCZGS2026] |
TTT-Discover |
| $1.502862$ |
[T2026] |
TogetherAI |
Known lower bounds
| Bound |
Reference |
Comments |
| $1$ |
Trivial |
|
| $1.182778$ |
[MO2004] |
|
| $1.262$ |
[MO2009] |
|
| $1.2748$ |
[MV2009] |
|
| $1.28$ |
[CS2017] |
|
| $1.2802$ |
[XX2026] |
Unpublished improvement, Grok |
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
- [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
- [XX2026] Xie, Xinyuan. Unpublished improvement to the lower bound for $C_{1a}$ (claiming $C_{1a} \ge 1.2802$). 2026. See Grok chat.
- [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