Green's Open Problem 28

Suppose that X, Y are two finitely-supported independent random variables taking integer values, and such that X + Y is uniformly distributed on its range. Are X and Y themselves uniformly distributed on their ranges?

Current state (tracked automatically)

Formal Conjectures marks the Lean statement green_28 as open.

All tracked facts and sources

Fact	Value	Source
Formal Conjectures: status of `green_28`	research open	Formal Conjectures
Formal Conjectures: research statements in the file	1 open, 0 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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-green-28,
  title        = {Green's Open Problem 28},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/green-28}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

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On the literature board

Nobody has worked on this problem here yet

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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

Suppose that $X, Y$ are two finitely-supported independent random variables taking integer values, and such that $X + Y$ is uniformly distributed on its range. Are $X$ and $Y$ themselves uniformly distributed on their ranges?

References:

Green, Ben. "100 open problems." (2024).
Mathoverflow/339137 asked by user Sil
MathStackexchange/3325163 asked by user Emmanuel Amiot

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.GreensOpenProblems.«28».

theorem green_28 :
  ∀ (X Y : PMF ℤ), -- marginals, independence is built into indepSum
    X.support.Finite ∧ Y.support.Finite ∧ IsUniformOnSupport (indepSum X Y) →
      IsUniformOnSupport X ∧ IsUniformOnSupport Y

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
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.

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.