44.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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

6 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 44: Extending Sidon Sets

References:

erdosproblems.com/44
[Si38] Singer, James, A theorem in finite projective geometry and some applications to number theory. Trans. Amer. Math. Soc. (1938), 377--385.

Imports (1)

FormalConjecturesUtil

theorem Erdos44.maxSidonSubsetCard_icc_bound The maximum size of a Sidon set in {1, ..., N} is less than or equal to 2 * √N line 38 source
Source documentation
The maximum size of a Sidon set in {1, ..., N} is less than or equal to 2 * √N.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Real, Finset.maxSidonSubsetCard, Finset.Icc, HMul.hMul, Real.sqrt
Proof uses: HPow.hPow, Finset, And, Membership.mem, Finset.filter, IsSidon, Finset.powerset, Eq, Finset.sup, Finset.card, Exists, Finset.exists_mem_eq_sup, Exists.intro, EmptyCollection.emptyCollection, of_eq_true, Eq.trans, HAdd.hAdd, Or, True, congrFun' and 90 more
157 instances, coercions and auxiliary constants left out.

Tactic trace: 49 steps
Open to load the checked before-and-after goal states.
```
theorem maxSidonSubsetCard_icc_bound (N : ℕ) (hN : 1 ≤ N) :
    maxSidonSubsetCard (Icc 1 N) ≤ 2 * Real.sqrt N := by
…
```
theorem Erdos44.erdos_44 Erdős Problem 44: Let N ≥ 1 and A ⊆ {1,…,N} be a Sidon set sorry line 92 source
Source documentation
Erdős Problem 44: Let N ≥ 1 and A ⊆ {1,…,N} be a Sidon set. Is it true that, for any ε > 0, there exist M = M(ε) and B ⊆ {N+1,…,M} such that A ∪ B ⊆ {1,…,M} is a Sidon set of size at least (1−ε)M^{1/2}?
This problem asks whether any Sidon set can be extended to achieve a density arbitrarily close to the optimal density for Sidon sets.

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Finset, LE.le, Finset.Icc, IsSidon, Real, GT.gt, Exists, And, HAdd.hAdd, Union.union, Set, HMul.hMul, HSub.hSub, Real.sqrt, Finset.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
56 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_44 : answer(sorry) ↔ ∀ᵉ (N ≥ (1 : ℕ)) (A ⊆ Finset.Icc 1 N), IsSidon (A : Set ℕ) →
    ∀ᵉ (ε > (0 : ℝ)), ∃ᵉ (M > N) (B ⊆ Finset.Icc (N + 1) M),
      IsSidon (A ∪ B : Set ℕ) ∧ (1 - ε) * Real.sqrt M ≤ (A ∪ B).card := by
…
```
Mentions: IsSidon
theorem Erdos44.erdos_44.variants.empty_start The case where we start with an empty set (constructing large Sidon sets) sorry line 106 source
Source documentation
The case where we start with an empty set (constructing large Sidon sets).
The answer is yes. Singer [Si38] constructed Sidon sets of size $q + 1$ in $\{1, \ldots, q^2 + q + 1\}$ for every prime $q$. Since $p_{n+1}/p_n \to 1$ for consecutive primes, this gives Sidon sets of size $(1 - o(1))M^{1/2}$ in $\{1, \ldots, M\}$ for every $M$; see erdosproblems.com/30.

L4DEL explanation
The case where we start with an empty set (constructing large Sidon sets). The answer is yes. Singer [Si38] constructed Sidon sets of size $q + 1$ in $\{1, \ldots, q^2 + q + 1\}$ for every prime $q$. Since $p_{n+1}/p_n \to 1$ for consecutive primes, this gives Sidon sets of size $(1 - o(1))M^{1/2}$ in $\{1, \ldots, M\}$ for every $M$; see erdosproblems.com/30.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, True, Real, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, Exists, Finset, And, LE.le, Finset.Icc, IsSidon, HMul.hMul, HSub.hSub, Real.sqrt, Finset.card, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_44.variants.empty_start : answer(True) ↔ ∀ᵉ (ε > (0 : ℝ)), ∀ᶠ (M : ℕ) in Filter.atTop,
    ∃ᵉ (A ⊆ Finset.Icc 1 M), IsSidon (A : Set ℕ) ∧ (1 - ε) * Real.sqrt M ≤ A.card := by
…
```
Mentions: IsSidon
theorem Erdos44.example_sidon_set The set {1, 2, 4, 8, 13} is a Sidon set in {1, ..., 13} line 116 source
Source documentation
The set {1, 2, 4, 8, 13} is a Sidon set in {1, ..., 13}.

standard axioms: propext what this means
Statement uses: IsSidon, Nat, Insert.insert, Set, OfNat.ofNat, Singleton.singleton
Proof uses: Membership.mem, Eq, HAdd.hAdd, Or, And, Eq.mp, Eq.trans, congrArg, of_eq_true, True, congr, eq_self, and_self, or_self, Eq.symm, False.elim, False, eq_false_of_decide, Eq.refl, Bool and 3 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
```
theorem example_sidon_set : IsSidon ({1, 2, 4, 8, 13} : Set ℕ) := by
…
```
Mentions: IsSidon
theorem Erdos44.sidon_set_lower_bound For any N, there exists a Sidon set of size at least √N/2 sorry line 129 source
Source documentation
For any N, there exists a Sidon set of size at least √N/2.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, Finset, And, Finset.Icc, IsSidon, Real, HDiv.hDiv, Real.sqrt, 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 sidon_set_lower_bound (N : ℕ) (hN : 1 ≤ N) :
    ∃ᵉ (A ⊆ Finset.Icc 1 N), IsSidon (A : Set ℕ) ∧ Real.sqrt N / 2 ≤ A.card := by
…
```
Mentions: IsSidon
theorem Erdos44.greedy_sidon_construction The greedy construction gives a Sidon set A ⊆ {1, ..., N} of size at least $N^{1/3}$, stated here as $N \le |A|^3$ sorry line 138 source
Source documentation
The greedy construction gives a Sidon set A ⊆ {1, ..., N} of size at least $N^{1/3}$, stated here as $N \le |A|^3$. See Section 1 of arXiv:2103.15850.

L4DEL explanation
The greedy construction gives a Sidon set A ⊆ {1, ..., N} of size at least $N^{1/3}$, stated here as $N \le |A|^3$. See Section 1 of arXiv:2103.15850.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, Finset, And, Finset.Icc, IsSidon, HPow.hPow, Finset.card
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem greedy_sidon_construction (N : ℕ) (hN : 1 ≤ N) :
    ∃ᵉ (A ⊆ Finset.Icc 1 N), IsSidon (A : Set ℕ) ∧ N ≤ A.card ^ 3 := by
…
```
Mentions: IsSidon