erdos_997.variants.irrational

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].

theorem erdos_997.variants.irrational :
    ∃ α : ℝ, Irrational α ∧
      ¬ IsWellDistributed (fun n ↦ Int.fract (α * (n.nth Nat.Prime)))

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