Mahler’s Conjecture on ξ(3/2)ⁿmod 1

K. Mahler’s conjecture: There exists no ξ ∈ ℝ⁺ such that the fractional parts {ξ(3/2)ⁿ} satisfy 0 ≤ {ξ(3/2)ⁿ} < 1/2 for all n = 0, 1, 2,... Such a ξ, if exists, is called a Mahler’s Z-number. In this paper we prove that if ξ is a Z-number, then the sequence x_n = {ξ(3/2)ⁿ}, n =1, 2,... has asymptotic distribution function c₀(x), where c₀(x)=1 for x ∈ (0, 1].