Let
$\unicode[STIX]{x1D6F9}:[1,\infty )\rightarrow \mathbb{R}_{+}$
be a non-decreasing function,
$a_{n}(x)$
the
$n$
th partial quotient of
$x$
and
$q_{n}(x)$
the denominator of the
$n$
th convergent. The set of
$\unicode[STIX]{x1D6F9}$
-Dirichlet non-improvable numbers,
$$\begin{eqnarray}G(\unicode[STIX]{x1D6F9}):=\{x\in [0,1):a_{n}(x)a_{n+1}(x)>\unicode[STIX]{x1D6F9}(q_{n}(x))\text{ for infinitely many }n\in \mathbb{N}\},\end{eqnarray}$$
is related with the classical set of
$1/q^{2}\unicode[STIX]{x1D6F9}(q)$
-approximable numbers
${\mathcal{K}}(\unicode[STIX]{x1D6F9})$
in the sense that
${\mathcal{K}}(3\unicode[STIX]{x1D6F9})\subset G(\unicode[STIX]{x1D6F9})$
. Both of these sets enjoy the same
$s$
-dimensional Hausdorff measure criterion for
$s\in (0,1)$
. We prove that the set
$G(\unicode[STIX]{x1D6F9})\setminus {\mathcal{K}}(3\unicode[STIX]{x1D6F9})$
is uncountable by proving that its Hausdorff dimension is the same as that for the sets
${\mathcal{K}}(\unicode[STIX]{x1D6F9})$
and
$G(\unicode[STIX]{x1D6F9})$
. 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].