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module
public import Mathlib.Algebra.Order.Archimedean.Real.Basic
public import Mathlib.Algebra.Order.Floor.Defs@[expose] public sectionMahler's Z-numbers
A Z-number is a positive real number ξ such that the fractional part of
ξ (3/2)^n lies in [0, 1/2) for every positive integer n. Mahler asked
whether any Z-number exists; this is still open.
References:
[Mah68] Mahler, Kurt. "An unsolved problem on the powers of 3/2." Journal of the Australian Mathematical Society 8.2 (1968): 313-321.
Main definitions
IsZNumber: a real number is a Z-number.
A Z-number is a positive real number ξ such that the fractional part of
ξ (3/2)^n lies in [0, 1/2) for every positive integer n.
def IsZNumber (ξ : ℝ) : Prop :=
0 < ξ ∧ ∀ n : ℕ, 1 ≤ n → Int.fract (ξ * (3 / 2 : ℝ) ^ n) < 1 / 2