Erdős Problem #1192 — open problem in combinatorics

Erdős Problem #1192

Does there exist, for all r≥ 2, a basis A of order r (so that f_r(n)>0 for all large n) such that Σ_n≤ xf_r(n)^2 ≪ x for all x?

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.

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Cite

@misc{cairn-erdos-1192,
  title        = {Erdős Problem #1192},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1192}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Does there exist, for all $r \geq 2$ , a basis $A$ of order $r$ (so that $f_{r} (n) > 0$ for all large $n$ ) such that $n \leq x \sum f_{r} (n)^{2} ≪ x$ for all $x$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1192». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1192 :
    answer(sorry) ↔
      ∀ r ≥ 2, ∃ A : Set ℕ,
        (∀ᶠ n in atTop, f_r A r n > 0) ∧
        (fun (x : ℕ) ↦ ∑ n ∈ range (x + 1), (f_r A r n : ℝ) ^ 2) =O[atTop]
          (fun (x : ℕ) ↦ (x : 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/1192. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1192
[Ru90] Ruzsa, Imre Z., A just basis. Monatsh. Math. (1990), 145--151.
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. 6 (1980), 89--115.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.