1180.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 1180

References:

erdosproblems.com/1180
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[Sh02] Shparlinski, Igor E., On a question of Erdős and Graham. Arch. Math. (Basel) (2002), 445-448.
[Gl06] Glibichuk, A. A., Combinatorial properties of sets of residues modulo a prime and the Erdős-Graham problem. Mat. Zametki (2006), 384-395.

Imports (1)

FormalConjecturesUtil

def Erdos1180.Represents Represents ε p a s says that the multiset s of denominators n with 1 ≤ n ≤ p^ε and n coprime to p has ∑ n⁻¹ = a in ZMod p line 43 source
Source documentation
Represents ε p a s says that the multiset s of denominators n with 1 ≤ n ≤ p^ε and n coprime to p has ∑ n⁻¹ = a in ZMod p.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat, ZMod, Multiset
Proof uses: And, Membership.mem, LE.le, OfNat.ofNat, HPow.hPow, Nat.Coprime, Eq, Multiset.sum, Multiset.map, Inv.inv
17 instances, coercions and auxiliary constants left out.
```
def Represents (ε : ℝ) (p : ℕ) (a : ZMod p) (s : Multiset ℕ) : Prop :=
…
```
def Erdos1180.C C ε is the least C such that, for all primes p, every residue modulo p is the sum of at most C elements of {n⁻¹ : 1 ≤ n ≤ p^ε} (or 0 if no such C exists) line 51 source
Source documentation
C ε is the least C such that, for all primes p, every residue modulo p is the sum of at most C elements of {n⁻¹ : 1 ≤ n ≤ p^ε} (or 0 if no such C exists).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Nat
Proof uses: InfSet.sInf, Set.ofPred, Nat.Prime, ZMod, Exists, Multiset, And, LE.le, Multiset.card, Erdos1180.Represents
2 instances, coercions and auxiliary constants left out.
```
noncomputable def C (ε : ℝ) : ℕ :=
…
```
theorem Erdos1180.erdos_1180 sorry line 69 source
depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Exists, Nat, Nat.Prime, ZMod, Multiset, And, LE.le, Multiset.card, Erdos1180.Represents
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos1180.erdos_1180.variants.shparlinski Shparlinski [Sh02] proved that $\ll \epsilon^{-3}$ summands suffice for every residue modulo every sufficiently large prime, where the threshold on $p$ may depend on $\epsilon$ sorry line 80 source
Source documentation
Shparlinski [Sh02] proved that $\ll \epsilon^{-3}$ summands suffice for every residue modulo every sufficiently large prime, where the threshold on $p$ may depend on $\epsilon$. No bound polynomial in $\epsilon^{-1}$ can hold for all primes, since when $p^\epsilon < 2$ the only admissible denominator is $1$.

depends on sorry what this means
Statement uses: Exists, Real, LT.lt, OfNat.ofNat, LE.le, Nat, Nat.Prime, ZMod, Multiset, And, Multiset.card, HMul.hMul, HPow.hPow, Inv.inv, Erdos1180.Represents
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_1180.variants.shparlinski : ∃ K : ℝ, ∀ ε : ℝ, 0 < ε → ε ≤ 1 →
    ∃ P : ℕ, ∀ p : ℕ, P ≤ p → p.Prime → ∀ a : ZMod p,
      ∃ s : Multiset ℕ, (s.card : ℝ) ≤ K * ε⁻¹ ^ 3 ∧ Represents ε p a s := by
…
```
Mentions: Erdos1180.Represents
theorem Erdos1180.erdos_1180.variants.glibichuk Glibichuk [Gl06] improved the bound to $\ll \epsilon^{-2}$ summands, again for every sufficiently large prime sorry line 90 source
Source documentation
Glibichuk [Gl06] improved the bound to $\ll \epsilon^{-2}$ summands, again for every sufficiently large prime.

depends on sorry what this means
Statement uses: Exists, Real, LT.lt, OfNat.ofNat, LE.le, Nat, Nat.Prime, ZMod, Multiset, And, Multiset.card, HMul.hMul, HPow.hPow, Inv.inv, Erdos1180.Represents
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_1180.variants.glibichuk : ∃ K : ℝ, ∀ ε : ℝ, 0 < ε → ε ≤ 1 →
    ∃ P : ℕ, ∀ p : ℕ, P ≤ p → p.Prime → ∀ a : ZMod p,
      ∃ s : Multiset ℕ, (s.card : ℝ) ≤ K * ε⁻¹ ^ 2 ∧ Represents ε p a s := by
…
```
Mentions: Erdos1180.Represents
theorem Erdos1180.erdos_1180.variants.lower_bound It is trivial that at least $\gg \epsilon^{-1}$ summands are required sorry line 97 source
Source documentation
It is trivial that at least $\gg \epsilon^{-1}$ summands are required.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, LE.le, HMul.hMul, Inv.inv, Erdos1180.C
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1180.variants.lower_bound : ∃ K : ℝ, 0 < K ∧ ∀ ε : ℝ, 0 < ε → ε ≤ 1 →
    K * ε⁻¹ ≤ C ε := by
…
```
Mentions: Erdos1180.C
theorem Erdos1180.erdos_1180.variants.optimal Is $C_\epsilon\leq \epsilon^{-1-o(1)}$? sorry line 103 source
Source documentation
Is $C_\epsilon\leq \epsilon^{-1-o(1)}$?

depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Filter.Eventually, LE.le, Erdos1180.C, HPow.hPow, Inv.inv, HAdd.hAdd, nhdsWithin, Set.Ioi
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
35 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1180.variants.optimal : answer(sorry) ↔ ∀ δ : ℝ, 0 < δ →
    ∀ᶠ ε : ℝ in nhdsWithin 0 (Set.Ioi 0), (C ε : ℝ) ≤ ε⁻¹ ^ (1 + δ) := by
…
```
Mentions: Erdos1180.C