FormalConjectures.ErdosProblems.«1188»

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import FormalConjecturesUtil

Erdős Problem 1188

Reference: erdosproblems.com/1188

namespace Erdos1188open Filteropen scoped Topology

A congruence class (n, a) represents a (mod n).

abbrev CongruenceClass := ℕ × ℕ

The modulus is > 1 and the residue is reduced.

def ValidClass (c : CongruenceClass) : Prop := 2 ≤ c.1 ∧ c.2 < c.1

z lies in the class (n, a).

def Satisfies (z : ℤ) (c : CongruenceClass) : Prop := z % (c.1 : ℤ) = (c.2 : ℤ)

Every integer satisfies some congruence of S.

def Covers (S : Finset CongruenceClass) : Prop := ∀ z : ℤ, ∃ c ∈ S, Satisfies z c

No two classes of S share a modulus.

def HasDistinctModuli (S : Finset CongruenceClass) : Prop :=
  ∀ ⦃c₁⦄, c₁ ∈ S → ∀ ⦃c₂⦄, c₂ ∈ S → c₁.1 = c₂.1 → c₁ = c₂

A minimal distinct covering system: valid classes with distinct moduli that cover ℤ, with no proper subset already covering.

def IsMinimalDistinctCoveringSystem (S : Finset CongruenceClass) : Prop :=
  (∀ c ∈ S, ValidClass c) ∧ HasDistinctModuli S ∧ Covers S ∧
    ∀ T : Finset CongruenceClass, T ⊂ S → ¬ Covers T

All classes (n, a) with 2 ≤ n ≤ x and a < n.

def ClassesUpTo (x : ℕ) : Finset CongruenceClass :=
  (Finset.Icc 2 x).biUnion fun n => (Finset.range n).image fun a => (n, a)

open scoped Classical in

F(x): the number of minimal distinct covering systems all of whose moduli lie in [1, x].

noncomputable def coveringCount (x : ℕ) : ℕ :=
  ((ClassesUpTo x).powerset.filter IsMinimalDistinctCoveringSystem).card

Call a set of distinct integers $1<n_1<\cdots<n_k$ with associated congruence classes $a_i\pmod{n_i}$ a distinct covering system if every integer satisfies at least one of these congruences. A minimal distinct covering system is one such that no proper subset forms a covering system. Let $F(x)$ count the number of minimal distinct covering systems with all moduli in $[1,x]$. Estimate $F(x)$.

The estimate is log(log F(x)) / log x → 1.

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-1188/Research/SparseAsymptotic.lean"]
theorem erdos_1188 :
    Tendsto (fun x : ℕ => Real.log (Real.log (coveringCount x : ℝ)) / Real.log (x : ℝ))
      atTop (𝓝 1) := by⊢ Tendsto (fun x ↦ Real.log (Real.log ↑(coveringCount x)) / Real.log ↑x) atTop (𝓝 1)
  sorryAll goals completed! 🐙

end Erdos1188