The sets of Dirichlet non-improvable numbers versus well-approximable numbers

Let Psi :[1, infinity) -> R+ be a non-decreasing function, a(n) (x) the nth partial quotient of x and q(n) (x) the denominator of the nth convergent. The set of psi-Dirichlet non-improvable numbers, G(Psi): = {x is an element of [0, 1) : a(n)(x)a(n+1) (x) > Psi(q(n)(x)) > Psi(q(n)(x)) for infinitely many n is an element of N], is related with the classical set of 1=q(2) Psi(q)-approximable numbers K(Psi) in the sense that K (3 Psi) subset of G(Psi) Both of these sets enjoy the same s-dimensional Hausdorff measure criterion for s is an element of (0, 1). We prove that the set G(Psi) \ K (3 Psi) is uncountable by proving that its Hausdorff dimension is the same as that for the sets K (Psi) and G (Psi). This gives an affirmative answer to a question raised by Hussain et al [Hausdorff measure of sets of Dirichlet non-improvable numbers. Mathematika 64(2) (2018), 502-518].