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

• Registered predicted interval (heuristic, unverified): $C_{1a}\in [1.497138480933,1.502861497147]$* by [S2026]. Dual-extremum structural estimate (phase-flipped crossover): lower edge $3/2-3\sqrt{6{\Phi }^2}\cdot \delta$, upper edge $3/2+3\sqrt{5\pi }\cdot \delta$, where $\delta =6{\Phi }^2-5\pi \approx 2.4066455\times 10^{-4}$ and $\Phi =(1+\sqrt{5})/2$. Preregistered kill criteria: any certified construction below $1.497138480933$ kills the interval; any certified lower bound above $1.502861497147$ kills the interval. Heuristic prediction from an unverified algebraic framework — not a certified construction, not a certified bound. All recorded construction values to date, including [A2026] $1.5027434868$, fall within the predicted interval.
• 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.
• On the registered predicted interval [S2026]: the fissure $\delta =6{\Phi }^2-5\pi$ carries a phase reading on the standard Fourier clock. Since ${\Phi }^2-5\pi /6=\delta /6\approx 4.01\times 10^{-5}$, the golden iteration unit ${\Phi }^2$, read as a phase angle, lands within $4\times 10^{-5}$ radians of $5\pi /6$ — the fifth hour of the twelve-fold division of the unit circle. Twelve is the crystallographic order $\mathrm{lcm}(2,3,4,6)$: the twelfth roots of unity (the discrete skeleton of the Fourier basis) contain every rotation symmetry allowed by the crystallographic restriction theorem, and $\sin (5\pi /6)=1/2$ is the self-dual mirror height. Equivalently, the pair $(6,5)$ is the second convergent $5/6$ of the continued fraction of ${\Phi }^2/\pi =[0;1,5,2175,2,\dots ]$ (parallel comment on the $C_{1b}$ page). The Gibbs overshoot constant $\approx 0.08949$ has continued fraction $[0;11,5,1,2,1,2,1,3,1]$ with no corresponding structure; no numerical Gibbs claim is made.
• Determinacy evidence (V3, doi:10.5281/zenodo.23603555, Appendix A–D). The registered interval above is carried by a full determinacy evidence chain, not by a free-parameter guess: its four endpoint formulas are completely determined by a single input $\Pi =4{\int }_0^{1}dx/(1+x^2)$ (the final factorization leaves $\pi$ as the only transcendental), and every certified judgement path strips $\pi$ by a Riemann-sum ladder (Appendix A/B): the decision $L_0({\pi }_k^{+})\le s\le U_0({\pi }_k^{-})$ evaluates only finite rational arithmetic — no transcendental arithmetic is ever evaluated — and the monotone convergence ${\pi }_k\to \pi$ fixes the judgement window to the claimed band width $\approx 5.7\times 10^{-3}$ at sufficiently large $k$. Independent anchor: the Atris construction 1.5027434867095834$\ast$ has been recomputed by us in pure-integer exact arithmetic (zero discrepancy) and falls inside the interval, $\approx 0.49\delta$ below the upper edge ($\delta =6{\Phi }^2-5\pi \approx 2.4066\times 10^{-4}$). Under the framework's power-reduction theorem ([0,1] decimal field $\equiv$ integer field; $\pi$ power-reduction), no transcendental arithmetic enters any certified judgement path, so finite-precision recomputation converges to the endpoint formulas themselves. Status under the repository's asterisk convention: still starred — lifting the asterisk requires replay by an independent checker, not by the submitting team's own verifier.
• Fingerprint disclosure. Pre-registered golden-signature fingerprint probes (height-level ratios, large-jump positions, spectral probes at $\phi$ and $\Phi$, and denominator-reduction checks) were run against the best-known 65,536-level step construction [A2026], with all judgment criteria fixed before the data was examined. All probes returned negative: the construction shape carries no golden-lattice quantization. This negative result withdraws only the shape-template interpretation of the golden anchor — the registered interval prediction above is unaffected, since its rationale (dual-extremum fixed-point balance plus the $O(\delta )$ fissure correction) is independent of construction shape. The kill lines are unchanged.

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.
• [A2026] Atris Labs. Record constructions for open math constants, with exact scorers, 2026. https://github.com/atrislabs/records/tree/1fa857a021ddbb4064708daf336758b866f75cb4
• [EA2026] EinsteinArena leaderboard, first autocorrelation inequality. https://einsteinarena.com/problems/first-autocorrelation-inequality
• [T2026b] Toledo, Carlos. Independent replay of the [A2026] and [EA2026] step functions for $C_{1a}$ and $C_{1b}$: the heights as exact rationals over one common denominator, the autoconvolution ($C_{1a}$) and the correlation with $1-h$ ($C_{1b}$) as exact integer products, and the maximum over every grid shift, in two implementations that share no code with each other or with the claimants' scorers. Report (§2); code and record at commit 5de570c (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 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.