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

Let be a non-decreasing function, the the partial quotient of and the denominator of the the convergent. The set of -Dirichlet non-improvable numbers, \unicode[STIX]{x1D6F9}(q_{n}(x))\text{ for infinitely many }n\in \mathbb{N}\},\end{eqnarray}]]> is related with the classical set of -approximable numbers in the sense that. Both of these sets enjoy the same -dimensional Hausdorff measure criterion for. We prove that the set is uncountable by proving that its Hausdorff dimension is the same as that for the sets and. This gives an affirmative answer to a question raised by Hussain etA al [Hausdorff measure of sets of Dirichlet non-improvable numbers. MathematikaA 64(2) (2018), 502-518].