Published online by Cambridge University Press: 07 December 2020
In this article we aim to investigate the Hausdorff dimension of the set of points
$x \in [0,1)$
such that for any
$r\in \mathbb {N}$
,
$n\in \mathbb {N}$
, where h and
$\tau $
are positive continuous functions, T is the Gauss map and
$a_{n}(x)$
denotes the nth partial quotient of x in its continued fraction expansion. By appropriate choices of
$r,\tau (x)$
and
$h(x)$
we obtain various classical results including the famous Jarník–Besicovitch theorem.
Communicated by Dzmitry Badziahin
This research was supported by a La Trobe University Postgraduate Research Award.