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Erdős problem 973

Does there exist a constant C>1C>1 such that, for every n≥2n\geq 2, there exists a sequence zi∈Cz_i\in \mathbb{C} with z1=1z_1=1 and ∣zi∣≥1\lvert z_i\rvert \geq 1 for all 1≤i≤n1\leq i\leq n with max⁡2≤k≤n+1∣∑1≤i≤nzik∣<C−n\max_{2\leq k\leq n+1}\left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert < C^{-n}?

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FormalConjectures/ErdosProblems/

973.lean

Retained formal statement1 of 4

Does there exist a constant C>1C>1 such that, for every n≥2n\geq 2, there exists a sequence zi∈Cz_i\in \mathbb{C} with z1=1z_1=1 and ∣zi∣≥1\lvert z_i\rvert \geq 1 for all 1≤i≤n1\leq i\leq n with max⁡2≤k≤n+1∣∑1≤i≤nzik∣<C−n\max_{2\leq k\leq n+1}\left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert < C^{-n}?

This is Problem 7.3 in [Ha74], where it is attributed to Erdős.

FormalConjectures/ErdosProblems/973.leanErdos973.erdos_9736 linesExact file
True ↔  ∃ C > 1,    ∀ n ≥ 2,      ∃ z,        z 1 = 1 ∧          (∀ i ∈ Finset.Icc 1 n, 1 ≤ ‖z i‖) ∧ ∀ k ∈ Finset.Icc 2 (n + 1), ‖∑ i ∈ Finset.Icc 1 n, z i ^ k‖ < C ^ (-↑n)
OpenStatement only, no proof

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