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import FormalConjecturesUtilErdős Problem 997
[APSSV26] B. Alexeev, M. Putterman, M. Sawhney, M. Sellke, and G. Valiant, Short proofs in combinatorics and number theory. arXiv:2603.29961 (2026).
[CLLW24] J. Champagne, T. Le, Y.-R. Liu, and T. D. Wooley, Well-distribution modulo one and the primes. arXiv:2406.19491 (2024).
[Er64b] Erdős, P., Problems and results on diophantine approximations. Compositio Math. (1964), 52-65.
[Er85e] Erdős, P., Some problems and results in number theory. Number theory and combinatorics. Japan 1984 (Tokyo, Okayama and Kyoto, 1984) (1985), 65-87.
[Hl55] Hlawka, Edmund, Zur formalen {T}heorie der {G}leichverteilung in kompakten {G}ruppen. Rend. Circ. Mat. Palermo (2) (1955), 33--47.
[Mo26] P. Monticone, Lean formalisation of Erdős problem 997 (2026)
open Set
namespace Erdos997
Call $x_1,x_2,\ldots \in (0,1)$ well-distributed if, for every $\epsilon>0$, if $k$ is sufficiently large then, for all $n>0$ and intervals $I\subseteq [0,1]$, $\lvert # { n < m\leq n+k : x_m\in I} - \lvert I\rvert k\rvert < \epsilon k.$
The notion of a well-distributed sequence was introduced by Hlawka and Petersen [Hl55].
def IsWellDistributed (x : ℕ → ℝ) : Prop :=
∀ ε > 0, ∀ᶠ k in Filter.atTop, ∀ n : ℕ,
∀ a b, 0 ≤ a → a ≤ b → b ≤ 1 →
letI I := Ico a b
let count := (Finset.Ioc n (n + k)).filter (fun m ↦ x m ∈ I)
abs ((count.card : ℝ) - (b - a) * k) < ε * k
Is it true that, for every $\alpha$, the sequence ${ \alpha p_n}$ is not well-distributed, if $p_n$ is the sequence of primes?
The answer is yes, by [APSSV26, Section 4]; a Lean formalisation is available in [Mo26].
@[category research solved, AMS 11, formal_proof using lean4 at
"https://live.lean-lang.org/#project=mathlib-v4.28.0&url=https://gist.githubusercontent.com/pitmonticone/016f2ed66b4cd1c4c4b9998095170e60/raw/b7dfc05c525ae385b5835f89f1ada721443e4305/Erdos997.lean"]
theorem erdos_997 :
answer(True) ↔
∀ α : ℝ, ¬ IsWellDistributed (fun n ↦ Int.fract (α * (n.nth Nat.Prime))) := ⊢ True ↔ ∀ (α : ℝ), ¬IsWellDistributed fun n => Int.fract (α * ↑(Nat.nth Nat.Prime n))
All goals completed! 🐙
Erdős proved that, if $n_k$ is a lacunary sequence, then the sequence ${ \alpha n_k}$ is not well-distributed for almost all $\alpha$.
@[category research solved, AMS 11]
theorem erdos_997.variants.lacunary (n : ℕ → ℕ) (h : IsLacunary n) :
∀ᵐ α, ¬ IsWellDistributed (fun k ↦ Int.fract (α * (n k : ℝ))) := n:ℕ → ℕh:IsLacunary n⊢ ∀ᵐ (α : ℝ), ¬IsWellDistributed fun k => Int.fract (α * ↑(n k))
All goals completed! 🐙
He also claimed in [Er64b] to have proved that there exists an irrational $\alpha$ for which ${\alpha p_n}$ is not well-distributed. He later retracted this claim in [Er85e], saying "The theorem is no doubt correct and perhaps will not be difficult to prove but I never was able to reconstruct my 'proof' which perhaps never existed."
The existence of such an $\alpha$ was established by Champagne, Le, Liu, and Wooley [CLLW24].
@[category research solved, AMS 11]
theorem erdos_997.variants.irrational :
∃ α : ℝ, Irrational α ∧
¬ IsWellDistributed (fun n ↦ Int.fract (α * (n.nth Nat.Prime))) := ⊢ ∃ α, Irrational α ∧ ¬IsWellDistributed fun n => Int.fract (α * ↑(Nat.nth Nat.Prime n))
All goals completed! 🐙
end Erdos997