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import FormalConjecturesUtilErdős Problem 450
Reference: erdosproblems.com/450
open Filteropen scoped Topologynamespace Erdos450
m has a divisor strictly between n and 2n.
def HasMediumDivisor (n m : ℕ) : Prop := ∃ d : ℕ, n < d ∧ d < 2 * n ∧ d ∣ mopen scoped Classical in
The number of integers strictly between x and x + y with a divisor in
(n, 2n).
noncomputable def localCount (n x y : ℕ) : ℕ :=
((Finset.Ioo x (x + y)).filter (HasMediumDivisor n)).card
Every window (x, x+y) has at most ε y integers with a divisor in (n, 2n).
def UniformlySparse (ε : ℝ) (n y : ℕ) : Prop := ∀ x : ℕ, (localCount n x y : ℝ) ≤ ε * (y : ℝ)
Y ε n is a sufficient window length: for every ε > 0, all large n, and
every y ≥ Y ε n, the window is ε-sparse.
def IsSufficientScale (Y : ℝ → ℕ → ℕ) : Prop :=
∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ n : ℕ, N ≤ n → ∀ y : ℕ, Y ε n ≤ y → UniformlySparse ε n yHow large must $y=y(\epsilon,n)$ be such that the number of integers in $(x,x+y)$ with a divisor in $(n,2n)$ is at most $\epsilon y$?
A linear scale is known to suffice (see erdos_450.linear_scale_suffices).
Whether the optimal scale is sublinear — a sufficient Y with Y ε n = o(n) —
is open.
@[category research open, AMS 11]
theorem erdos_450 : answer(sorry) ↔
∃ Y : ℝ → ℕ → ℕ, IsSufficientScale Y ∧
∀ ε : ℝ, 0 < ε → Tendsto (fun n : ℕ => (Y ε n : ℝ) / n) atTop (𝓝 0) := ⊢ True ↔ ∃ Y, IsSufficientScale Y ∧ ∀ (ε : ℝ), 0 < ε → Tendsto (fun n ↦ ↑(Y ε n) / ↑n) atTop (𝓝 0)
All goals completed! 🐙
A translate-uniform linear scale suffices: there is a sufficient window
length Y with Y ε n ≤ C(ε) · n. This is an upper bound on the optimal scale,
not the exact threshold asked for in erdos_450.
@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-450/Research/TuranAnswer.lean"]
theorem erdos_450.linear_scale_suffices :
∃ Y : ℝ → ℕ → ℕ,
(∀ ε : ℝ, 0 < ε → ∃ C : ℝ, ∀ n : ℕ, (Y ε n : ℝ) ≤ C * n) ∧ IsSufficientScale Y := ⊢ ∃ Y, (∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (n : ℕ), ↑(Y ε n) ≤ C * ↑n) ∧ IsSufficientScale Y
All goals completed! 🐙end Erdos450