40.lean`

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4 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 40

Reference: erdosproblems.com/40

Imports (2)

def Erdos40.Erdos40For The predicate for a function $g\colon\mathbb{N} → \mathbb{R})$ that $$\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}$$ implies $\limsup 1_A\ast 1_A(n)=\infty$ line 39 source
Source documentation
The predicate for a function $g\colon\mathbb{N} → \mathbb{R})$ that $$\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}$$ implies $\limsup 1_A\ast 1_A(n)=\infty$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real
Proof uses: Set, Asymptotics.IsBigO, Filter.atTop, HDiv.hDiv, Real.sqrt, Set.ncard, Inter.inter, Set.Icc, OfNat.ofNat, Eq, ENat, Filter.limsup, AdditiveCombinatorics.sumRep, Top.top
15 instances, coercions and auxiliary constants left out.
```
def Erdos40For (g : ℕ → ℝ) : Prop :=
…
```
theorem Erdos40.erdos_40 For what functions $g(N) → \infty$ is it true that $$\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}$$ implies $\limsup 1_A\ast 1_A(n)=\infty$? sorry line 50 source
Source documentation
For what functions $g(N) → \infty$ is it true that $$\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}$$ implies $\limsup 1_A\ast 1_A(n)=\infty$?

depends on sorry what this means
Statement uses: Eq, Set, Nat, Real, Set.ofPred, And, Filter.Tendsto, Filter.atTop, Erdos40.Erdos40For, sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_40 :
    {g : ℕ → ℝ | Tendsto g atTop atTop ∧ Erdos40For g} = answer(sorry) := by
…
```
Mentions: Erdos40.Erdos40For
theorem Erdos40.erdos_40.variants.weaker Is there any function $g(N) → \infty$ such that $$\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}$$ implies $\limsup 1_A\ast 1_A(n)=\infty$? sorry line 65 source
Source documentation
Is there any function $g(N) → \infty$ such that $$\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}$$ implies $\limsup 1_A\ast 1_A(n)=\infty$?
This is a weaker form of Erdős Problem 40, which asks for all such $g$. Establishing the implication for even one $g(N) → \infty$ already answers Erdős Problem 28 positively, because a basis of order $2$ satisfies $\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2}$.

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Real, And, Filter.Tendsto, Filter.atTop, Erdos40.Erdos40For
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_40.variants.weaker :
    answer(sorry) ↔ ∃ g : ℕ → ℝ, Tendsto g atTop atTop ∧ Erdos40For g := by
…
```
Mentions: Erdos40.Erdos40For
theorem Erdos40.erdos_40.variants.implies_erdos_28 Establishing the property in Erdős Problem 40 for any one function $g(N) → \infty$ implies the Erdős-Turán conjecture, see Erdős Problem 28 line 74 source
Source documentation
Establishing the property in Erdős Problem 40 for any one function $g(N) → \infty$ implies the Erdős-Turán conjecture, see Erdős Problem 28.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Filter.Tendsto, Filter.atTop, Erdos40.Erdos40For, Set, Set.Finite, Compl.compl, HAdd.hAdd, Eq, ENat, Filter.limsup, AdditiveCombinatorics.sumRep, Top.top
Proof uses: Membership.mem, LE.le, Exists, Asymptotics.IsBigO, HDiv.hDiv, Real.sqrt, Set.ncard, Inter.inter, Set.Icc, OfNat.ofNat, Set.Finite.exists_le, Asymptotics.IsBigO.of_bound, Filter.mp_mem, Set.ofPred, Norm.norm, HMul.hMul, Filter.Tendsto.eventually_ge_atTop, Filter.eventually_ge_atTop, Filter.univ_mem', id and 144 more
210 instances, coercions and auxiliary constants left out.

Tactic trace: 52 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_40.variants.implies_erdos_28 (g : ℕ → ℝ) (hg : Tendsto g atTop atTop)
    (h_erdos_40 : Erdos40For g) : type_of% Erdos28.erdos_28 := by
…
```
Mentions: Erdos40.Erdos40For, Erdos28.erdos_28