337.lean`

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

4 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 337

References:

erdosproblems.com/337
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[ErGr80b] Erdős, P. and Graham, R. L., On bases with an exact order. Acta Arith. (1980), 201-207.
[RT85] Ruzsa, I. Z. and Turjányi, S., A note on additive bases of integers. Publ. Math. Debrecen (1985), 101-104.
[Tu84] Turjányi, S., A note on basis sequences. Topics in classical number theory, Vol. I, II (Budapest, 1981) (1984), 1571-1576.

Imports (1)

FormalConjecturesUtil

theorem Erdos337.erdos_337 Let $A\subseteq \mathbb{N}$ be an additive basis (of any finite order) such that $\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$ sorry line 64 source
Source documentation
Let $A\subseteq \mathbb{N}$ be an additive basis (of any finite order) such that $\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$. Is it true that $$ \lim_{N\to \infty}\frac{\lvert (A+A)\cap \{1,\ldots,N\}\rvert} {\lvert A\cap \{1,\ldots,N\}\rvert}=\infty? $$
The answer is no, and a counterexample was provided by Turjányi [Tu84]. This was generalised (to the replacement of $A+A$ by the $h$-fold sumset $hA$ for any $h\geq 2$) by Ruzsa and Turjányi [RT85].
"Additive basis" is Set.IsAsymptoticAddBasis: some finite $h$ has $hA$ containing every sufficiently large integer. The exact notion Set.IsAddBasis, which asks that $hA$ be all of $\mathbb{N}$, would force $0, 1 \in A$ and is not the class these results are about.
The linked file states the basis hypothesis as ∃ N₀, Set.Ici N₀ ⊆ iterated_sumset A k and indexes both counting functions by a real $x$ through $\lfloor x\rfloor$, where the counting functions here are indexed by $N : \mathbb{N}$.

depends on sorry what this means
Statement uses: Iff, False, Set, Nat, Set.IsAsymptoticAddBasis, Asymptotics.IsLittleO, Real, Filter.atTop, Set.ncard, Inter.inter, Set.Icc, OfNat.ofNat, Filter.Tendsto, HDiv.hDiv, HAdd.hAdd, Set.add
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_337 : answer(False) ↔
    ∀ A : Set ℕ, A.IsAsymptoticAddBasis →
      (fun N : ℕ ↦ ((A ∩ Icc 1 N).ncard : ℝ)) =o[atTop] (fun N : ℕ ↦ (N : ℝ)) →
      Tendsto (fun N : ℕ ↦ (((A + A) ∩ Icc 1 N).ncard : ℝ) / ((A ∩ Icc 1 N).ncard : ℝ))
        atTop atTop := by
…
```
theorem Erdos337.erdos_337.variants.h_fold This was generalised (to the replacement of $A+A$ by the $h$-fold sumset $hA$ for any $h\geq 2$) by Ruzsa and Turjányi [RT85] sorry line 76 source
Source documentation
This was generalised (to the replacement of $A+A$ by the $h$-fold sumset $hA$ for any $h\geq 2$) by Ruzsa and Turjányi [RT85].

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, Set, And, Set.IsAsymptoticAddBasis, Asymptotics.IsLittleO, Real, Filter.atTop, Set.ncard, Inter.inter, Set.Icc, Not, Filter.Tendsto, HDiv.hDiv, HSMul.hSMul, Set.NSMul
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_337.variants.h_fold : ∀ h : ℕ, 2 ≤ h →
    ∃ A : Set ℕ, A.IsAsymptoticAddBasis ∧
      (fun N : ℕ ↦ ((A ∩ Icc 1 N).ncard : ℝ)) =o[atTop] (fun N : ℕ ↦ (N : ℝ)) ∧
      ¬ Tendsto (fun N : ℕ ↦
          ((h • A ∩ Icc 1 N).ncard : ℝ) / ((A ∩ Icc 1 N).ncard : ℝ))
        atTop atTop := by
…
```
theorem Erdos337.erdos_337.variants.ruzsa_turjanyi sorry line 99 source
depends on sorry what this means
Statement uses: Set, Nat, Set.IsAsymptoticAddBasis, Asymptotics.IsLittleO, Real, Filter.atTop, Set.ncard, Inter.inter, Set.Icc, OfNat.ofNat, Filter.Tendsto, HDiv.hDiv, HAdd.hAdd, Set.add, HMul.hMul
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 Erdos337.erdos_337.variants.three_fold sorry line 120 source
depends on sorry what this means
Statement uses: Set, Nat, Set.IsAsymptoticAddBasis, Asymptotics.IsLittleO, Real, Filter.atTop, Set.ncard, Inter.inter, Set.Icc, OfNat.ofNat, Filter.Tendsto, HDiv.hDiv, HAdd.hAdd, Set.add, HMul.hMul
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.