Ben Green's Open Problem 35

Lower bound for c(p) for 1 < p ≤ ∞, improving the known value √(4/7) at p = 2 or the known value 0.64 at p = ∞.

Current state (tracked automatically)

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: research statements in the file	2 open, 5 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-green-35,
  title        = {Ben Green's Open Problem 35},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-35}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

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

green_35.upper. Upper bound for $c (p)$ for $1 < p \leq \infty$ , 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).

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

theorem green_35.upper :
    (∀ p, 1 < p → c p ≤ green_35.upper.answer p) ∧ green_35.upper.answer ∞ < 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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.