1193.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 1193

References:

erdosproblems.com/1193
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.

Imports (1)

FormalConjecturesUtil

theorem Erdos1193.erdos_1193.parts.i Let $A\subset \mathbb{N}$ and let $g(n)$ be a non-decreasing function of $n$ which is always $>0$ sorry line 47 source
Source documentation
Let $A\subset \mathbb{N}$ and let $g(n)$ be a non-decreasing function of $n$ which is always $>0$.
Is the lower density of $$\{ n : 1_A\ast 1_A(n)=g(n)\}$$ always $0$?
The answer is trivially no to both questions: indeed if $A=\mathbb{N}$ (assuming $0\in\mathbb{N}$) then $1_A\ast 1_A(n)=n+1$ for all $n$. Presumably Erdős had some additional restrictions on either $g$ or $A$ in mind, but these are not recorded in [Er80].

depends on sorry what this means
Statement uses: Iff, False, Set, Nat, Monotone, LT.lt, OfNat.ofNat, Eq, Real, Set.lowerDensity, Set.ofPred, AdditiveCombinatorics.sumRep, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1193.parts.i : answer(False) ↔
    ∀ (A : Set ℕ) (g : ℕ → ℕ), Monotone g → (∀ n, 0 < g n) →
      {n : ℕ | sumRep A n = g n}.lowerDensity = 0 := by
…
```
theorem Erdos1193.erdos_1193.parts.ii Let $A\subset \mathbb{N}$ and let $g(n)$ be a non-decreasing function of $n$ which is always $>0$ sorry line 64 source
Source documentation
Let $A\subset \mathbb{N}$ and let $g(n)$ be a non-decreasing function of $n$ which is always $>0$.
Is the upper density of $$\{ n : 1_A\ast 1_A(n)=g(n)\}$$ always $<c$ for some constant $c<1$?
The answer is trivially no to both questions: indeed if $A=\mathbb{N}$ (assuming $0\in\mathbb{N}$) then $1_A\ast 1_A(n)=n+1$ for all $n$. Presumably Erdős had some additional restrictions on either $g$ or $A$ in mind, but these are not recorded in [Er80].

depends on sorry what this means
Statement uses: Iff, False, Exists, Real, And, LT.lt, OfNat.ofNat, Set, Nat, Monotone, Set.upperDensity, Set.ofPred, Eq, AdditiveCombinatorics.sumRep, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1193.parts.ii : answer(False) ↔
    ∃ c < (1 : ℝ), ∀ (A : Set ℕ) (g : ℕ → ℕ), Monotone g → (∀ n, 0 < g n) →
      {n : ℕ | sumRep A n = g n}.upperDensity < c := by
…
```
Mentions: c
theorem Erdos1193.erdos_1193.variants.sumRep_univ Indeed if $A=\mathbb{N}$ (assuming $0\in\mathbb{N}$) then $1_A\ast 1_A(n)=n+1$ for all $n$ sorry line 73 source
Source documentation
Indeed if $A=\mathbb{N}$ (assuming $0\in\mathbb{N}$) then $1_A\ast 1_A(n)=n+1$ for all $n$.

depends on sorry what this means
Statement uses: Nat, Eq, AdditiveCombinatorics.sumRep, Set.univ, HAdd.hAdd, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1193.variants.sumRep_univ (n : ℕ) : sumRep (Set.univ : Set ℕ) n = n + 1 := by
…
```
theorem Erdos1193.erdos_1193.variants.upper_density_pos Erdős writes the upper density can be positive, but he believes it is bounded away from $1$ sorry line 80 source
Source documentation
Erdős writes the upper density can be positive, but he believes it is bounded away from $1$.

depends on sorry what this means
Statement uses: Exists, Set, Nat, And, Monotone, LT.lt, OfNat.ofNat, Real, Set.upperDensity, Set.ofPred, Eq, AdditiveCombinatorics.sumRep, Set.univ
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1193.variants.upper_density_pos :
    ∃ (A : Set ℕ) (g : ℕ → ℕ), Monotone g ∧ (∀ n, 0 < g n) ∧
      0 < {n : ℕ | sumRep A n = g n}.upperDensity := by
…
```