Erdős problem 987 · problems.science

Clunie [Cl67] proved that there exists an infinite sequence $\{z_\nu\}$ on the unit circle with $A_\nu \le \nu$ for all $\nu \ge 1$ . Translating $z_\nu = e(x_\nu)$ , the natural domain of $x_\nu$ is the half-open unit interval $\mathrm{Ico}\,0\,1$ , matching the original [Er64b]/[Cl67] statement (any unit complex number is allowed, including $z = 1$ , i.e. $x = 0$ ).

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10 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

987.lean

Retained formal statement3 of 10

Liu [Li69] showed that, for any $\epsilon > 0$ , $A_k \gg k^{1 - \epsilon}$ infinitely often under the additional assumption that there are only a finite number of distinct points. Clunie observed in the Mathscinet review of [Li69] that under this assumption in fact $A_k = \infty$ infinitely often (the version stated here).

FormalConjectures/ErdosProblems/987.leanErdos987.erdos_987.variants.finite_distinct_points1 lineExact file

1∀ (x : ℕ → ℝ), (∀ (j : ℕ), x j ∈ Set.Ioo 0 1) → (Set.range x).Finite → ∃ᶠ (k : ℕ) in Filter.atTop, Erdos987.A x k = ⊤

SolvedProof has a holelean4external proof

The proof uses `sorry`: part of the argument is written but not proved. Lean accepts the file; it does not accept the theorem.

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