867.lean`

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

7 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 867

References:

erdosproblems.com/867
[CoPh96] Coppersmith, Don and Phillips, Steven, On a question of Erdős on subsequence sums. SIAM J. Discrete Math. (1996), 173-177.
[Fr93] Freud, R., Adding numbers - on a problem of P. Erdős. James Cook Mathematical Notes (1993), 6199-6202.

Imports (1)

FormalConjecturesUtil

def Erdos867.ConsecutiveSumFree A finite set of naturals line 41 source
Source documentation
A finite set of naturals $A=\{a_1<\cdots<a_t\}$ is consecutive-sum-free if it has no solutions to $a_i+a_{i+1}+\cdots+a_j\in A$ with $i<j$; equivalently, whenever an interval $[m,n]$ contains at least two elements of $A$, the sum of the elements of $A$ lying in $[m,n]$ is not itself an element of $A$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Nat
Proof uses: LE.le, OfNat.ofNat, Finset.card, Inter.inter, Finset.Icc, Not, Membership.mem, Finset.sum
9 instances, coercions and auxiliary constants left out.
```
def ConsecutiveSumFree (A : Finset ℕ) : Prop :=
…
```
theorem Erdos867.erdos_867 Documented declaration: erdos_867 sorry line 56 source
Source documentation
Is it true that if $A=\{a_1<\cdots <a_t\}\subseteq \{1,\ldots,N\}$ has no solutions to $$a_i+a_{i+1}+\cdots+a_j\in A$$ then $$\lvert A\rvert \leq \frac{N}{2}+O(1)?$$
In fact this problem is false. Freud [Fr93] constructed a sequence with density $\geq 19/36$. The current best bounds are due to Coppersmith and Phillips [CoPh96], who prove that the maximal size of such an $A$ satisfies $$\frac{13}{24}N -O(1)\leq \lvert A\rvert \leq \left(\frac{2}{3}-\frac{1}{512}\right)N+\log N.$$

depends on sorry what this means
Statement uses: Iff, False, Exists, Real, Nat, Finset, LE.le, Finset.Icc, OfNat.ofNat, Erdos867.ConsecutiveSumFree, Finset.card, HAdd.hAdd, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_867 : answer(False) ↔
    ∃ C : ℝ, ∀ N : ℕ, ∀ A ⊆ Finset.Icc 1 N, ConsecutiveSumFree A →
      (A.card : ℝ) ≤ (N : ℝ) / 2 + C := by
…
```
Mentions: Erdos867.ConsecutiveSumFree
theorem Erdos867.erdos_867.variants.lower_bound Taking $A=(N/2,N]\cap \mathbb{N}$ shows $\lvert A\rvert \geq N/2-O(1)$ is possible line 65 source
Source documentation
Taking $A=(N/2,N]\cap \mathbb{N}$ shows $\lvert A\rvert \geq N/2-O(1)$ is possible.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Exists, Real, Nat, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, Erdos867.ConsecutiveSumFree, HSub.hSub, HDiv.hDiv, Finset.card
Proof uses: Exists.intro, HAdd.hAdd, And.intro, Membership.mem, Eq.mpr, id, Eq, Decidable.byContradiction, Not, Eq.mp, Inter.inter, Finset.sum, rfl, Eq.trans, congrFun', congrArg, congr, HMul.hMul, mul_comm, HSMul.hSMul and 78 more
138 instances, coercions and auxiliary constants left out.

Tactic trace: 28 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_867.variants.lower_bound :
    ∃ C : ℝ, ∀ N : ℕ, ∃ A ⊆ Finset.Icc 1 N, ConsecutiveSumFree A ∧
      ((N : ℝ) / 2 - C ≤ (A.card : ℝ)) := by
…
```
Mentions: Erdos867.ConsecutiveSumFree
theorem Erdos867.erdos_867.variants.adenwalla Adenwalla has observed that $$\lvert A\rvert \leq (\tfrac{2}{3}+o(1))N.$$ sorry line 98 source
Source documentation
Adenwalla has observed that $$\lvert A\rvert \leq (\tfrac{2}{3}+o(1))N.$$

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, Finset, LE.le, Finset.Icc, Erdos867.ConsecutiveSumFree, Finset.card, HMul.hMul, HAdd.hAdd, HDiv.hDiv, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_867.variants.adenwalla (ε : ℝ) (hε : 0 < ε) :
    ∀ᶠ N : ℕ in atTop, ∀ A ⊆ Finset.Icc 1 N, ConsecutiveSumFree A →
      (A.card : ℝ) ≤ (2 / 3 + ε) * (N : ℝ) := by
…
```
Mentions: Erdos867.ConsecutiveSumFree
theorem Erdos867.erdos_867.variants.freud Freud [Fr93] constructed a sequence with density $\geq 19/36$ sorry line 107 source
Source documentation
Freud [Fr93] constructed a sequence with density $\geq 19/36$.

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, Exists, Finset, And, LE.le, Finset.Icc, Erdos867.ConsecutiveSumFree, HMul.hMul, HSub.hSub, HDiv.hDiv, Finset.card, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
44 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_867.variants.freud :
    ∀ ε : ℝ, 0 < ε → ∀ᶠ N : ℕ in atTop, ∃ A ⊆ Finset.Icc 1 N, ConsecutiveSumFree A ∧
      (((19 / 36 : ℝ) - ε) * (N : ℝ) ≤ (A.card : ℝ)) := by
…
```
Mentions: Erdos867.ConsecutiveSumFree
theorem Erdos867.erdos_867.variants.coppersmith_phillips_lower_bound The current best bounds are due to Coppersmith and Phillips [CoPh96], who prove that the maximal size of such an $A$ satisfies $$\frac{13}{24}N -O(1)\leq \lvert A\rvert.$$ sorry line 118 source
Source documentation
The current best bounds are due to Coppersmith and Phillips [CoPh96], who prove that the maximal size of such an $A$ satisfies $$\frac{13}{24}N -O(1)\leq \lvert A\rvert.$$

depends on sorry what this means
Statement uses: Exists, Real, Nat, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, Erdos867.ConsecutiveSumFree, HSub.hSub, HMul.hMul, HDiv.hDiv, Finset.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_867.variants.coppersmith_phillips_lower_bound :
    ∃ C : ℝ, ∀ N : ℕ, ∃ A ⊆ Finset.Icc 1 N, ConsecutiveSumFree A ∧
      ((13 / 24 : ℝ) * (N : ℝ) - C ≤ (A.card : ℝ)) := by
…
```
Mentions: Erdos867.ConsecutiveSumFree
theorem Erdos867.erdos_867.variants.coppersmith_phillips_upper_bound Documented declaration: coppersmith_phillips_upper_bound sorry line 129 source
Source documentation
The current best bounds are due to Coppersmith and Phillips [CoPh96], who prove that the maximal size of such an $A$ satisfies $$\lvert A\rvert \leq \left(\frac{2}{3}-\frac{1}{512}\right)N+\log N.$$

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, Finset, LE.le, Finset.Icc, OfNat.ofNat, Erdos867.ConsecutiveSumFree, Real, Finset.card, HAdd.hAdd, HMul.hMul, HSub.hSub, HDiv.hDiv, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
46 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_867.variants.coppersmith_phillips_upper_bound :
    ∀ᶠ N : ℕ in atTop, ∀ A ⊆ Finset.Icc 1 N, ConsecutiveSumFree A →
      (A.card : ℝ) ≤ (2 / 3 - 1 / 512) * (N : ℝ) + Real.log (N : ℝ) := by
…
```
Mentions: Erdos867.ConsecutiveSumFree