/-
Copyright 2026 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
https://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/
import FormalConjecturesUtilErdős Problem 973
[Er92f] Erdős, L., On some problems of {P}. Turán concerning power sums of complex numbers. Acta Math. Hungar. (1992), 11--24.
[Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.
[Tu84b] Turán, Paul, On a new method of analysis and its applications. (1984), xvi+584.
open Finset Filter
namespace Erdos973
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 $\max_{2\leq k\leq n+1}\left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert < C^{-n}$?
This is Problem 7.3 in [Ha74], where it is attributed to Erdős.
@[category research open, AMS 11]
theorem erdos_973 :
answer(sorry) ↔
∃ C : ℝ, C > 1 ∧
∀ n : ℕ, n ≥ 2 → ∃ z : ℕ → ℂ,
z 1 = 1 ∧
(∀ i ∈ Icc 1 n, 1 ≤ ‖z i‖) ∧
(∀ k ∈ Icc 2 (n + 1), ‖∑ i ∈ Icc 1 n, z i ^ k‖ < C ^ (-(n : ℝ))) := ⊢ True ↔
∃ C > 1,
∀ n ≥ 2, ∃ z, z 1 = 1 ∧ (∀ i ∈ Icc 1 n, 1 ≤ ‖z i‖) ∧ ∀ k ∈ Icc 2 (n + 1), ‖∑ i ∈ Icc 1 n, z i ^ k‖ < C ^ (-↑n)
All goals completed! 🐙
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 $C\approx 1.32$.
@[category research solved, AMS 11]
theorem erdos_973.variants.le_one :
∃ C : ℝ, C > 1 ∧
∀ n : ℕ, n ≥ 2 → ∃ z : ℕ → ℂ,
z 1 = 1 ∧
(∀ i ∈ Icc 1 n, ‖z i‖ ≤ 1) ∧
(∀ k ∈ Icc 2 (n + 1), ‖∑ i ∈ Icc 1 n, z i ^ k‖ < C ^ (-(n : ℝ))) := ⊢ ∃ C > 1, ∀ n ≥ 2, ∃ z, z 1 = 1 ∧ (∀ i ∈ Icc 1 n, ‖z i‖ ≤ 1) ∧ ∀ k ∈ Icc 2 (n + 1), ‖∑ i ∈ Icc 1 n, z i ^ k‖ < C ^ (-↑n)
All goals completed! 🐙
In [Er92f] (a different) Erdős refines this analysis, proving that if $M_2=\min_{z_j} \max_{2\leq k\leq n+1} \left\lvert \sum_{1\leq j\leq n}z_j^k\right\rvert$ where the minimum is taken over all $z_j\in \mathbb{C}$ with $\max \lvert z_j\rvert=1$, then $(1.746)^{-n} < M_2 < (1.745)^{-n}$.
@[category research solved, AMS 11]
theorem erdos_973.variants.m2_bounds :
∀ n : ℕ, n ≥ 2 → ∀ M_2 : ℝ,
IsGLB { M | ∃ z : ℕ → ℂ,
(∀ j ∈ Icc 1 n, ‖z j‖ ≤ 1) ∧
(∃ j ∈ Icc 1 n, ‖z j‖ = 1) ∧
(∃ k ∈ Icc 2 (n + 1), M = ‖∑ j ∈ Icc 1 n, z j ^ k‖ ∧
∀ m ∈ Icc 2 (n + 1), ‖∑ j ∈ Icc 1 n, z j ^ m‖ ≤ M) } M_2 →
(1.746 : ℝ) ^ (-(n : ℝ)) < M_2 ∧ M_2 < (1.745 : ℝ) ^ (-(n : ℝ)) := ⊢ ∀ n ≥ 2,
∀ (M_2 : ℝ),
IsGLB
{M |
∃ z,
(∀ j ∈ Icc 1 n, ‖z j‖ ≤ 1) ∧
(∃ j ∈ Icc 1 n, ‖z j‖ = 1) ∧
∃ k ∈ Icc 2 (n + 1), M = ‖∑ j ∈ Icc 1 n, z j ^ k‖ ∧ ∀ m ∈ Icc 2 (n + 1), ‖∑ j ∈ Icc 1 n, z j ^ m‖ ≤ M}
M_2 →
1.746 ^ (-↑n) < M_2 ∧ M_2 < 1.745 ^ (-↑n)
All goals completed! 🐙
Tang notes in the comments that Theorem 6.1 of [Tu84b] implies that, if $\lvert z_i\rvert \geq 1$ for all $i$, then $\max_{2\leq k\leq n+1}\left\lvert \sum_{1\leq i\leq n}z_i^k\right\rvert \geq (2e)^{-(1+o(1))n}$.
@[category research solved, AMS 11]
theorem erdos_973.variants.tang :
∃ f : ℕ → ℝ, f =o[atTop] (fun _ ↦ (1 : ℝ)) ∧
∀ᶠ n in atTop, ∀ z : ℕ → ℂ,
(∀ i ∈ Icc 1 n, 1 ≤ ‖z i‖) →
∃ k ∈ Icc 2 (n + 1),
‖∑ i ∈ Icc 1 n, z i ^ k‖ ≥
(2 * Real.exp 1) ^ (-(1 + f n) * (n : ℝ)) := ⊢ ∃ f,
(f =o[atTop] fun x => 1) ∧
∀ᶠ (n : ℕ) in atTop,
∀ (z : ℕ → ℂ),
(∀ i ∈ Icc 1 n, 1 ≤ ‖z i‖) →
∃ k ∈ Icc 2 (n + 1), ‖∑ i ∈ Icc 1 n, z i ^ k‖ ≥ (2 * Real.exp 1) ^ (-(1 + f n) * ↑n)
All goals completed! 🐙
end Erdos973