221.lean`

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 221

References:

erdosproblems.com/221
[Lo54] Lorentz, G. G., On a problem of additive number theory. Proc. Amer. Math. Soc. (1954), 838-841.
[Ru72] Ruzsa, Jr., I., On a problem of P. Erdős. Canad. Math. Bull. (1972), 309-310.

Imports (1)

FormalConjecturesUtil

theorem Erdos221.erdos_221 Is there a set sorry line 46 source
Source documentation
Is there a set $A\subset\mathbb{N}$ such that, for all large $N$, $$\lvert A\cap\{1,\ldots,N\}\rvert \ll N/\log N$$ and such that every large integer can be written as $2^k+a$ for some $k\geq 0$ and $a\in A$?
Lorentz [Lo54] proved there is such a set with, for all large $N$, $$\lvert A\cap\{1,\ldots,N\}\rvert \ll \frac{\log\log N}{\log N}N$$ The answer is yes, proved by Ruzsa [Ru72].

depends on sorry what this means
Statement uses: Iff, True, Exists, Set, Nat, And, Asymptotics.IsBigO, Real, Filter.atTop, Set.ncard, Set.ofPred, Membership.mem, LE.le, HDiv.hDiv, Real.log, Filter.Eventually, OfNat.ofNat, Eq, HAdd.hAdd, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_221 :
    answer(True) ↔ ∃ A : Set ℕ,
      ((fun N => ({a ∈ A | a ≤ N}.ncard : ℝ)) ≪ (fun N => (N : ℝ) / Real.log N)) ∧
      ∀ᶠ N in atTop, ∃ k a, 0 ≤ k ∧ a ∈ A ∧ N = 2 ^ k + a := by
…
```