A remark on liminf sets in diophantine approximation

Let Q be an infinite set of positive integers, τ > 1 be a real number and let Wτ(Q)={x ∈ ℝ : |x-pq| < q-τ for infinitely many (p, q) ∈ ℤ × Q}.For any given positive integer m, set Q(m)={n ∈ ℕ : (n, m)=1}.Q(m)={n ∈ ℕ : (n, m)=1}.If m   is divisible by at least two prime factors, Adiceam [1] showed that Wτ(ℕ)\Wτ(Q(m))Wτ(ℕ)\Wτ(Q(m)) contains uncountably many Liouville numbers, and asked if it contains any non-Liouville numbers? In this note, we give an affirmative answer to Adiceam's question.