Sums of numbers of the form 2^k 3^l

DefinitionErdős Problem 845#✓fully proved

The proposition asserts that for every real number \(C > 0\), the set of all sums \(\sum_{(k, l) \in B} 2^k 3^l\), where \(B\) is a nonempty finite set of pairs of natural numbers such that \(\max_{(k, l) \in B} 2^k 3^l \le C \min_{(k, l) \in B} 2^k 3^l\), has natural density \(0\).

def conjecture : Prop :=
  ∀ (C : ℝ), 0 < C →
    let f : ℕ × ℕ → ℕ := fun (k, l) ↦ 2 ^ k * 3 ^ l
    { ∑ x ∈ B, f x | (B : Finset (ℕ × ℕ)) (h : B.Nonempty)
      (_hB : B.sup f ≤ C * B.inf' h f) }.HasDensity 0

Uses 1 · Used by 1

Statement usesDensity of a set

Used byErdős Problem 845 is false

TheoremErdős Problem 845 is false#●ready to prove

There is a real number \(C > 0\) such that the set of all sums \(\sum_{(k, l) \in B} 2^k 3^l\), where \(B\) is a nonempty finite set of pairs of natural numbers such that \(\max_{(k, l) \in B} 2^k 3^l \le C \min_{(k, l) \in B} 2^k 3^l\), does not have natural density \(0\).

Van Doorn and Everts obtain a counterexample with \(C = 6\).

public theorem_wanted conjecture_refuted : ¬ conjecture

Sources

• Wouter van Doorn and Anneroos R. F. Everts, Smooth sums with small spacings, arXiv:2511.04585v1 (2025), https://arxiv.org/abs/2511.04585v1.

Uses 1

Statement usesErdős Problem 845

1 fully proved1 ready to prove

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