FormalConjectures.ErdosProblems.«333»

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import FormalConjecturesUtil

Erdős Problem 333

References:

erdosproblems.com/333

[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

[ErNe77] Erdős, P. and Newman, D. J., Bases for sets of integers. J. Number Theory (1977), 420-425.

open Filter Asymptoticsopen scoped Pointwisenamespace Erdos333

Let $A\subseteq \mathbb{N}$ be a set of density zero. Does there exist a $B$ such that $A\subseteq B+B$ and $$\lvert B\cap {1,\ldots,N}\rvert =o(N^{1/2})$$ for all large $N$?

The answer is no. Erdős and Newman [ErNe77] have proved this is true when $A$ is the set of squares. In fact, Theorem 2 of [ErNe77] already implies a negative answer to this problem, but this seems to have been overlooked by Erdős and Graham.

See also [806].

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos333.lean"]
theorem erdos_333 : answer(False) ↔
    ∀ A : Set ℕ, A.HasDensity 0 →
      ∃ B : Set ℕ, A ⊆ B + B ∧
        (fun N => ((B ∩ Set.Icc 1 N).ncard : ℝ)) =o[atTop]
          fun N => Real.sqrt N := by⊢ False ↔ ∀ (A : Set ℕ), A.HasDensity 0 → ∃ B, A ⊆ B + B ∧ (fun N ↦ ↑(B ∩ Set.Icc 1 N).ncard) =o[atTop] fun N ↦ √↑N
  sorryAll goals completed! 🐙

end Erdos333