Countable dense sets as preimages under entire functions

DefinitionErdős's problem on countable dense preimages#✓fully proved

This is the statement that for any two countable dense subsets \(A\) and \(B\) of \(\mathbb{C}\), there is an entire function \(f : \mathbb{C} \to \mathbb{C}\) such that for every \(z \in \mathbb{C}\), \(f(z) \in B\) if and only if \(z \in A\); that is, \(f^{-1}(B) = A\).

def conjecture : Prop :=
  ∀ A B : Set ℂ, A.Countable → Dense A → B.Countable → Dense B →
    ∃ f : ℂ → ℂ, DifferentiableOn ℂ f Set.univ ∧ ∀ z, (f z ∈ B ↔ z ∈ A)

Sources

• Research Problems in Function Theory, Problem 2.31 (record AMR-022-2031).

Used by 1

Used byEvery countable dense set is a preimage of any other under an entire function

TheoremEvery countable dense set is a preimage of any other under an entire function#●ready to prove

For any two countable dense subsets \(A\) and \(B\) of \(\mathbb{C}\), there is an entire function \(f : \mathbb{C} \to \mathbb{C}\) such that for every \(z \in \mathbb{C}\), \(f(z) \in B\) if and only if \(z \in A\); that is, \(f^{-1}(B) = A\).

Hayman's update to Problem 2.31 reports that Barth and Schneider constructed such functions.

public theorem_wanted conjecture_holds : conjecture

Sources

• K. F. Barth and W. J. Schneider, Entire functions mapping arbitrary countable dense sets and their complements onto each other, J. London Math. Soc. (2) 4 (1972), 482-488, https://doi.org/10.1112/jlms/s2-4.3.482.

Uses 1

Statement usesErdős's problem on countable dense preimages

1 fully proved1 ready to prove

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