I'll present a result, joint with Rao Li and Baowei Wang, in which we calculate the Hausdorff dimension of the set $$ \{x\in [0,1): a_{n+k}(x) \geq q_n^t(x) \hbox{ for infinitely many } n\in \mathbb N\} $$ where $a_n(x)$ is the continuous fraction expansion of $x$ and $q_n(x)$ is the $n$-th denominator of best rational approximation of $x$. For $k=1$ this is a classical Jarnik set, but for $k\geq 2$ there is an interesting phase transition between case $t>2$ and $t<2$.