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 |
| $1.5029$ |
[YKLBMWKCZGS2026] |
TTT-Discover |
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Unpublished improvement, Grok |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 Unpublished improvement to the lower bound for $C_{1a}$ (claiming $C_{1a} \ge 1.2802$). 2026.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.