An autocorrelation constant for Sidon sets

If a non-negative function on an interval has total mass 1, how small can the peak of its autoconvolution $f * f$ be? The answer is a constant between 1.2802 and about 1.50286; it controls how large generalized Sidon sets can be, and any upper-bound improvement is a step function that can be checked exactly.

Problem

Let $C$ be the largest constant such that

$$\max_{-1/2 \le t \le 1/2} \int_{\mathbb{R}} f(t-x)\, f(x)\, dx \;\ge\; C \left( \int_{-1/4}^{1/4} f(x)\, dx \right)^2$$

for every non-negative function $f \colon \mathbb{R} \to \mathbb{R}$.

Restricting $f$ to $[-1/4, 1/4]$ only lowers the left side, so equivalently $C$ is the infimum of $\max_t (f * f)(t) / \left(\int f\right)^2$ over non-negative $f$ supported on $[-1/4, 1/4]$, not identically zero. Determine $C$, or narrow the known range.

Background

The constant comes from additive combinatorics: lower bounds on $C$ turn into upper bounds on the size of generalized Sidon sets, sets of integers in which every number has only boundedly many representations as a sum of two elements. Schinzel and Schmidt (2002) gave the upper bound $\pi/2$ and Martin and O'Bryant studied the problem systematically; Cloninger and Steinerberger (2017) proved a lower bound by a computer-assisted argument and applied it to Sidon sets. Since 2025 the upper bound has become a benchmark for computer search, with a run of small improvements by several groups.

What is known

• Upper bounds come from explicit functions: $\pi/2$ (Schinzel and Schmidt, 2002); $1.50992$ (Matolcsi and Vinuesa, 2010); $1.5053$, then $1.503164$ (Georgiev, Gómez-Serrano, Tao and Wagner, 2025); $1.502870$ (Yuksekgonul et al., 2026); the current verified record is a step function published by Together AI in 2026, recorded on the source as $1.502862$.
• Unverified claims (marked with an asterisk on the source): $1.5027435$ from the EinsteinArena leaderboard (October 2026), refined to $1.5027434868$ by Atris Labs with a 65,536-step function with rational heights.
• Lower bounds need proofs: $1.182778$ (Martin and O'Bryant, 2007) and $1.262$ (Martin and O'Bryant, 2009), $1.2748$ (Matolcsi and Vinuesa, 2010), $1.28$ (Cloninger and Steinerberger, 2017) and $1.2802$ (Ivanisvili and Xie, 2026). A lower bound of $1.292$ (Piterbarg, Bajaj and Vincent, 2026) is unverified.

What would count as progress

• A non-negative step function on $[-1/4, 1/4]$ whose ratio is below the recorded $1.502862$, and ideally below the claimed $1.5027434868$. The autoconvolution of a step function with equal steps is piecewise linear, so its maximum sits at a breakpoint and the ratio can be computed exactly in rational arithmetic in seconds.
• An independent exact check of the unverified $1.50274$ constructions, which would turn them into accepted records.
• A proof that $C > 1.2802$, typically a computer-assisted argument whose certificate can be rerun, or a verification of the claimed $1.292$.
• The constant is stated in Lean in formal-conjectures, with the open bounds as targets.

References

1. M. Matolcsi and C. Vinuesa, "Improved bounds on the supremum of autoconvolutions", Journal of Mathematical Analysis and Applications 372 (2010), 439–447. doi:10.1016/j.jmaa.2010.07.030, arXiv:0907.1379
2. A. Cloninger and S. Steinerberger, "On suprema of autoconvolutions with an application to Sidon sets", Proceedings of the AMS 145 (2017), 3191–3200. doi:10.1090/proc/13690
3. G. Martin and K. O'Bryant, "The supremum of autoconvolutions, with applications to additive number theory", Illinois Journal of Mathematics 53 (2009), 219–235. doi:10.1215/ijm/1264170847, arXiv:0807.5121
4. B. Georgiev, J. Gómez-Serrano, T. Tao and A. Z. Wagner, "Mathematical exploration and discovery at scale" (2025). arXiv:2511.02864
5. P. Ivanisvili and X. Xie, "Grokability in five inequalities" (2026). arXiv:2605.05193

Source

Imported from Optimization Constants in Mathematics, constant 1a, curated by Terence Tao and contributors. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.