Ben Green's Open Problem 35

The question

green_35.lower. Lower bound for $c(p)$ for $1<p\le \infty$, improving the known value $\sqrt{4/7}$ at $p=2$ or the known value $0.64$ at $p=\infty$.

green_35.upper. Upper bound for $c(p)$ for $1<p\le \infty$, improving the best-known value $0.7516$ at $p=\infty$.

Estimate the infimum of the $L^p$ norm of the self-convolution of a nonnegative integrable function supported on $[0,1]$ with total integral $1$.

We model a function f : [0,1] → ℝ≥0 as a function f : ℝ → ℝ that is nonnegative, integrable, supported on [0,1], and has total integral 1.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«35» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem green_35.lower :
    let lb : ℝ≥0∞ → ℝ≥0∞ := answer(sorry)
    (∀ p, 1 < p → lb p ≤ c p) ∧
      (ENNReal.ofReal (Real.sqrt (4 / 7)) < lb 2 ∨ 0.64 < lb ∞)

theorem green_35.upper :
    let ub : ℝ≥0∞ → ℝ≥0∞ := answer(sorry)
    (∀ p, 1 < p → c p ≤ ub p) ∧ ub ∞ < 0.7516

What counts as progress

• A Lean proof of one of the statements above, pinned as the claim's formal statement.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

• Ben Green's Open Problem 35
• Gr01 B. J. Green, The number of squares and $B_h[g]$-sets, Acta Arith. 100 (2001), no. 4, 365-390.
• CS17 A. Cloninger and S. Steinerberger, *On suprema of autoconvolutions with an application to Sidon sets*, Proc. Amer. Math. Soc. 145 (2017), no. 8, 3191-3200.
• MV10 M. Matolcsi and C. Vinuesa, Improved bounds on the supremum of autoconvolutions, J. Math. Anal. Appl. 372 (2010), 439-447.
• AE25 A. Novikov et al., AlphaEvolve: A coding agent for scientific and algorithmic discovery, arXiv:2506.13131 (2025), Appendix B.1.
• GGTW25 B. Georgiev, J. Gómez-Serrano, T. Tao and A. Z. Wagner, *Mathematical exploration and discovery at scale*, arXiv:2511.02864 (2025), Section 6.2.

The constants of [CS17], [MV10], [AE25] and [GGTW25] are stated for functions supported on $[-1/4,1/4]$; rescaling to $[0,1]$ halves them.

Source and licence

Imported from Formal Conjectures (Ben Green's 100 open problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.