Erdős Problem #749
Let ε>0. Does there exist A⊆ ℕ such that the lower density of A+A is at least 1-ε and yet 1_Aast 1_A(n) ≪_ε 1 for all n?
Current state (tracked automatically)
- erdosproblems.com lists Erdős problem #749 as open (status updated 31 August 2025).
- Formal Conjectures marks the Lean statement
erdos_749as open.
All tracked facts and sources
| Fact | Value | Source |
|---|---|---|
| Status of Erdős problem #749 on erdosproblems.com | open 2025-08-31 | erdosproblems.com |
| Formal Conjectures: status of `erdos_749` | 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): erdosproblems.com (checked 2026-11-12), 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. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.
Cite
@misc{cairn-erdos-749,
title = {Erdős Problem #749},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/erdos-749}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/erdos-749)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Nobody has worked on this problem here yet
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- Get a task — one click
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- Paste its answer here — check it and publish it under your name
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.
ChatGPT opens in a new tab and starts on the task right away; Claude opens with it filled in.
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
Let . Does there exist such that the lower density of is at least and yet for all ?
Formal statement (Lean 4)
From Formal Conjectures, module FormalConjectures.ErdosProblems.«749».
theorem erdos_749 : ∀ ε > (0 : ℝ),
∃ A : Set ℕ, 1 - ε ≤ lowerDensity (A + A) ∧
((Nat.cast (R := ℝ) ∘ sumRep A) ≪ (fun n => (1: ℝ)))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 — check erdosproblems.com/749. Report it as a literature claim.
- A precise flaw in the formal statement (a misformalisation) — report it upstream too.
References
Source and licence
Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.