333.lean`

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L4DEL: Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

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.

Imports (1)

FormalConjecturesUtil

theorem Erdos333.erdos_333 Let $A\subseteq \mathbb{N}$ be a set of density zero sorry line 52 source
Source documentation
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].

depends on sorry what this means
Statement uses: Iff, False, Set, Nat, Set.HasDensity, OfNat.ofNat, Real, Set.univ, Exists, And, LE.le, HAdd.hAdd, Asymptotics.IsLittleO, Filter.atTop, Set.ncard, Inter.inter, Set.Icc, Real.sqrt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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
…
```