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 statement5 of 10

Linear upper bound (tight via Clunie phase tracking). Tighter than linear_upper_bound_weak: there exists a sequence $x \in (0,1)$ with $A_k \le k + 1$ for all $k \ge 1$ , via the (shifted) van der Corput sequence. Whether the $+1$ can be eliminated to recover Clunie's exact $A_k \le k$ under the strict $\mathrm{Ioo}\,0\,1$ hypothesis is open in this formalization (see linear_upper_bound).

FormalConjectures/ErdosProblems/987.leanErdos987.erdos_987.variants.linear_upper_bound_clunie1 lineExact file

1∃ x, ∃ (_ : ∀ (j : ℕ), x j ∈ Set.Ioo 0 1), ∀ (k : ℕ), 1 ≤ k → Erdos987.A x k ≤ ↑↑k + 1

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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