1. FormalConjectures
  2. ErdosProblems
  3. «226»
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Erdős Problem 226

References:

    erdosproblems.com/226

    [BaSc70] Barth, K. F. and Schneider, W. J., Entire functions mapping countable dense subsets of the reals onto each other monotonically. J. London Math. Soc. (2) (1970), 620--626.

    [BaSc71] Barth, K. F. and Schneider, W. J., Entire functions mapping arbitrary countable dense sets and their complements onto each other. J. London Math. Soc. (2) (1971/72), 482--488.

    [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155--180.

namespace Erdos226

A real function preserves rationality.

def PreservesRationality (f : ℝ → ℝ) : Prop := ∀ x : ℝ, x ∈ Set.range ((↑) : ℚ → ℝ) ↔ f x ∈ Set.range ((↑) : ℚ → ℝ)

Is there an entire non-linear function $f$ such that, for all $x\in\mathbb{R}$, $x$ is rational if and only if $f(x)$ is?

Barth and Schneider [BaSc70] proved the stronger result for countable dense subsets of $\mathbb{R}$.

@[category research solved, AMS 30, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos226.lean"] theorem declaration uses 'sorry'erdos_226 : answer(True) ↔ ∃ F : ℂ → ℂ, Differentiable ℂ F ∧ (∀ x : ℝ, (F x).im = 0) ∧ (∀ g : AffineMap ℝ ℝ ℝ, (fun x : ℝ => (F x).re) ≠ g) ∧ PreservesRationality (fun x : ℝ => (F x).re) := ⊢ True ↔ ∃ F, Differentiable ℂ F ∧ (∀ (x : ℝ), (F ↑x).im = 0) ∧ (∀ (g : ℝ →ᵃ[ℝ] ℝ), (fun x => (F ↑x).re) ≠ ⇑g) ∧ PreservesRationality fun x => (F ↑x).re All goals completed! 🐙 end Erdos226