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Erdős problem 973

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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 statement2 of 4

Erdős proved (as described on p.35 of [Tu84b]) that such a sequence does exist with ∣zi∣≤1\lvert z_i\rvert\leq 1. Indeed, Erdős' construction gives a value of C≈1.32C\approx 1.32.

FormalConjectures/ErdosProblems/973.leanErdos973.erdos_973.variants.le_one5 linesExact file
∃ C > 1,  ∀ n ≥ 2,    ∃ z,      z 1 = 1 ∧        (∀ i ∈ Finset.Icc 1 n, ‖z i‖ ≤ 1) ∧ ∀ k ∈ Finset.Icc 2 (n + 1), ‖∑ i ∈ Finset.Icc 1 n, z i ^ k‖ < C ^ (-↑n)
SolvedStatement only, no proof