FormalConjectures.ErdosProblems.«973»

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module

public import FormalConjecturesUtil

Erdős Problem 973

References:

erdosproblems.com/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.

[LYZ26] Luo, Yanping and Yang, Ruiyi and Zhu, Keheng, Exterior power sums. arxiv/2607.22017

[Tu84b] Turán, Paul, On a new method of analysis and its applications. (1984), xvi+584.

@[expose] public sectionopen Finset Filternamespace 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.

The answer is no, by Luo, Yang and Zhu [LYZ26]: the maximum exceeds $e^{-\lambda n}$ for every fixed $\lambda>0$ once $n$ is large, so it decays subexponentially and no such $C$ exists. See erdos_973.variants.luo_yang_zhu below.

@[category research solved, AMS 11]
theorem erdos_973 :
    answer(False) ↔
      ∃ 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))) := by⊢ False ↔
  ∃ 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)
  sorryAll goals completed! 🐙

The result of [LYZ26] that settles the problem: for every fixed $\lambda>0$ and all large enough $n$, any $z_1,\dots,z_n$ with $\lvert z_j\rvert\geq 1$ satisfy $\max_{2\leq k\leq n+1}\left\lvert\sum_j z_j^k\right\rvert > e^{-\lambda n}$.

This is sharper than erdos_973.variants.tang, whose bound $(2e)^{-(1+o(1))n}$ pins the rate at $\log(2e) = 1 + \log 2$, still exponential decay.

@[category research solved, AMS 11]
theorem erdos_973.variants.luo_yang_zhu (lam : ℝ) (hlam : 0 < lam) :
    ∀ᶠ n in atTop, ∀ z : ℕ → ℂ,
      (∀ i ∈ Icc 1 n, 1 ≤ ‖z i‖) →
      ∃ k ∈ Icc 2 (n + 1),
        Real.exp (-lam * n) < ‖∑ i ∈ Icc 1 n, z i ^ k‖ := bylam:ℝhlam:0 < lam⊢ ∀ᶠ (n : ℕ) in atTop,
  ∀ (z : ℕ → ℂ), (∀ i ∈ Icc 1 n, 1 ≤ ‖z i‖) → ∃ k ∈ Icc 2 (n + 1), Real.exp (-lam * ↑n) < ‖∑ i ∈ Icc 1 n, z i ^ k‖
  sorryAll 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))) := by⊢ ∃ 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)
  sorryAll 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 $(1.746)^{-n} < M_2 < (1.745)^{-n}$ for all sufficiently large $n$.

@[category research solved, AMS 11]
theorem erdos_973.variants.m2_bounds :
    ∀ᶠ n : ℕ in atTop, ∀ 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)) := by⊢ ∀ᶠ (n : ℕ) in atTop,
  ∀ (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)
  sorryAll 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)) := by⊢ ∃ 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)
  sorryAll goals completed! 🐙

end Erdos973