Turán's three-value problem

DefinitionMultiplicity of a root of \(f = w\)#✓fully proved

Let \(f : \mathbb{C} \to \mathbb{C}\) and \(w, z \in \mathbb{C}\). The multiplicity of \(z\) as a root of the equation \(f = w\) is an extended natural number: it is \(\infty\) if \(f(t) = w\) for every \(t \neq z\) in some neighbourhood of \(z\); otherwise it is \(n\) if \(f - w\) has meromorphic order \(n \ge 0\) at \(z\), and it is \(0\) if that order is negative or \(f - w\) is not meromorphic at \(z\).

It depends only on the values of \(f\) on a punctured neighbourhood of \(z\), so changing \(f(z)\) does not change it. It is \(0\) when \(f\) has a pole at \(z\), and when \(f - w\) is analytic at \(z\) with \(f(z) \neq w\).

noncomputable def rootMult (f : ℂ → ℂ) (w z : ℂ) : ℕ∞ :=
  if meromorphicOrderAt (fun t => f t - w) z = ⊤ then ⊤
  else ((meromorphicOrderAt (fun t => f t - w) z).untopD 0).toNat
Sources
Used by 1
Used byMultiplicity of a value on the Riemann sphere

DefinitionMultiplicity of a pole#✓fully proved

Let \(f : \mathbb{C} \to \mathbb{C}\) and \(z \in \mathbb{C}\). The multiplicity of \(z\) as a pole of \(f\), that is, the multiplicity of the value \(\infty\) at \(z\), is \(-n\) if \(f\) has finite meromorphic order \(n < 0\) at \(z\), and \(0\) otherwise. In particular it is \(0\) unless \(f\) has a pole at \(z\).

noncomputable def poleMult (f : ℂ → ℂ) (z : ℂ) : ℕ∞ :=
  ((-(meromorphicOrderAt f z).untopD 0).toNat : ℕ∞)
Sources
Used by 1
Used byMultiplicity of a value on the Riemann sphere

DefinitionMultiplicity of a value on the Riemann sphere#✓fully proved

Let \(f : \mathbb{C} \to \mathbb{C}\) and \(z \in \mathbb{C}\), and let \(w\) be a point of the Riemann sphere \(\mathbb{C} \cup \{\infty\}\). The multiplicity of \(w\) as a value of \(f\) at \(z\) is the multiplicity of \(z\) as a root of \(f = w\) when \(w\) is finite, and the multiplicity of \(z\) as a pole of \(f\) when \(w = \infty\).

noncomputable def rootMultSph (f : ℂ → ℂ) (w : Option ℂ) (z : ℂ) : ℕ∞ :=
  match w with
  | some w => rootMult f w z
  | none => poleMult f z
Sources
Uses 2 · Used by 2
Statement usesMultiplicity of a pole · Multiplicity of a root of \(f = w\)
Used byTurán's three-value question · Bounded value distribution

DefinitionBounded value distribution#✓fully proved

A function \(f : \mathbb{C} \to \mathbb{C}\) has bounded value distribution if for every real \(r > 0\) there is a natural number \(C\) such that, for every value \(w\) of the Riemann sphere \(\mathbb{C} \cup \{\infty\}\) and every open disc of radius \(r\), the multiplicities of \(w\) as a value of \(f\) at the points of any finite subset of the disc add up to at most \(C\).

The source states this for functions meromorphic in the plane; the definition itself does not assume meromorphy.

def IsBoundedValueDistribution (f : ℂ → ℂ) : Prop :=
  ∀ (r : ℝ), r > 0 →
    ∃ C : ℕ, ∀ (c : ℂ) (w : Option ℂ) (s : Finset ℂ),
      (↑s ⊆ Metric.ball c r) → ∑ z ∈ s, rootMultSph f w z ≤ (C : ℕ∞)
Sources
Uses 1 · Used by 1
Statement usesMultiplicity of a value on the Riemann sphere
Used byTurán's three-value question

DefinitionTurán's three-value question#✓fully proved

The statement that every function \(f : \mathbb{C} \to \mathbb{C}\) with the following properties has bounded value distribution: \(f\) is meromorphic on \(\mathbb{C}\); for every \(w \in \mathbb{C}\) there is a point at which \(f - w\) has finite meromorphic order; and there are three distinct values \(w_1, w_2, w_3\) of the Riemann sphere \(\mathbb{C} \cup \{\infty\}\) such that for every real \(r > 0\) there is a natural number \(C\) for which, for each \(i\) and every open disc of radius \(r\), the multiplicities of \(w_i\) as a value of \(f\) at the points of any finite subset of the disc add up to at most \(C\).

def conjecture : Prop :=
  ∀ (f : ℂ → ℂ), MeromorphicOn f Set.univ →
    (∀ w : ℂ, ∃ z : ℂ, meromorphicOrderAt (fun t => f t - w) z ≠ ⊤) →
    (∃ w₁ w₂ w₃ : Option ℂ, w₁ ≠ w₂ ∧ w₁ ≠ w₃ ∧ w₂ ≠ w₃ ∧
      ∀ (r : ℝ), r > 0 →
        ∃ C : ℕ, ∀ (c : ℂ) (w : Option ℂ), w ∈ ({w₁, w₂, w₃} : Set (Option ℂ)) →
          ∀ (s : Finset ℂ),
            (↑s ⊆ Metric.ball c r) → ∑ z ∈ s, rootMultSph f w z ≤ (C : ℕ∞)) →
    IsBoundedValueDistribution f
Sources
Uses 2 · Used by 1
Statement usesBounded value distribution · Multiplicity of a value on the Riemann sphere
Used byTurán's three-value question has a negative answer

TheoremTurán's three-value question has a negative answer#●ready to prove

There is a function \(f : \mathbb{C} \to \mathbb{C}\), meromorphic on \(\mathbb{C}\), such that for every \(w \in \mathbb{C}\) there is a point at which \(f - w\) has finite meromorphic order, together with three distinct values \(w_1, w_2, w_3\) of the Riemann sphere \(\mathbb{C} \cup \{\infty\}\) such that for every real \(r > 0\) there is a natural number \(C\) for which, for each \(i\) and every open disc of radius \(r\), the multiplicities of \(w_i\) as a value of \(f\) at the points of any finite subset of the disc add up to at most \(C\); and yet \(f\) does not have bounded value distribution.

A. A. Gol'dberg (Izv. Vyssh. Uchebn. Zaved. Mat. 1972) answered Turan's question negatively, as recorded in Hayman-Lingham Update 2.28; the Weierstrass sigma-function is another counterexample: it has boundedly many \(w\)-points per disc for every bounded set of \(w\) but has order \(2\), so it does not have bounded value distribution.

public theorem_wanted conjecture_refuted : ¬ conjecture
Sources
  • W. K. Hayman and E. F. Lingham, Research Problems in Function Theory (2018), arXiv:1809.07200, https://arxiv.org/abs/1809.07200.
Uses 1
Statement usesTurán's three-value question
5 fully proved1 ready to prove

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