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 statement9 of 10

Clunie [Cl67] proved that, for every infinite sequence $x_1, x_2, \ldots \in (0, 1)$ , $A_k \gg k^{1/2}$ for infinitely many $k$ . (Tao independently found a proof.)

FormalConjectures/ErdosProblems/987.leanErdos987.erdos_987.variants.sqrt_lower_bound1 lineExact file

1∀ (x : ℕ → ℝ), (∀ (j : ℕ), x j ∈ Set.Ioo 0 1) → ∃ c > 0, ∃ᶠ (k : ℕ) in Filter.atTop, ↑(c * √↑k) ≤ 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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