In 1938, Erdős and Mahler raised the following question:
Let $\xi$ be a real number such that $(p_n/q_n)_n$ are convergents of its continued fraction. If there exists a subsequence $(p_{n_j}/q_{n_j})_j$ of the convergents such that $P(p_{n_j}q_{n_j})\leq M$ for all $j$ (where $P(m)$ denotes the greatest prime factor of $m$), then $\xi$ is a Liouville number.
Erdős and Mahler proved the existence of numbers with this property.
I would like to ask you if it is possible to construct this subsequence with the property that $n_j-n_{j-1}\leq k$ for some constant $k$ and all $j$.
Thanks in advance