Erdős problem 973 · problems.science

Does there exist a constant $C>1$ such that, for every $n\geq 2$ , there exists a sequence $z_i\in \mathbb{C}$ with $z_1=1$ and $\lvert z_i\rvert \geq 1$ for all $1\leq i\leq n$ with $\max_{2\leq k\leq n+1}\left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert < C^{-n}$ ?

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4 retained statements2415f78e850a

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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 $\lvert z_i\rvert\leq 1$ . Indeed, Erdős' construction gives a value of $C\approx 1.32$ .

FormalConjectures/ErdosProblems/973.leanErdos973.erdos_973.variants.le_one5 linesExact file

1∃ C > 1,2  ∀ n ≥ 2,3    ∃ z,4      z 1 = 1 ∧5        (∀ 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

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