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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FormalConjectures/ErdosProblems/

987.lean

Retained formal statement10 of 10

Erdős [Er64b] remarks it is "easy to see" that for every infinite sequence $x_1, x_2, \ldots \in (0, 1)$ , $\limsup_{k \to \infty} \sup_n \left\lvert \sum_{j \le n} e(k x_j) \right\rvert = \infty.$

FormalConjectures/ErdosProblems/987.leanErdos987.erdos_987.variants.sup_limsup_infty3 linesExact file

1∀ (x : ℕ → ℝ),2  (∀ (j : ℕ), x j ∈ Set.Ioo 0 1) →3    Filter.limsup (fun k => ⨆ n, ↑‖∑ j ∈ Finset.range n, additiveChar (↑k * x j)‖) Filter.atTop = ⊤

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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