It is well known that if $0.a_1a_2a_3\dots$ is the base-$b$ expansion of a number normal to base-$b$, then the numbers $0.a_ka_{ m+k}a_{ 2m+k}\dots$ for $m\ge 2$, $k\ge 1$ are all normal to base-$b$ as well. In contrast, given a continued fraction expansion $\langle a_1,a_2,a_3,\dots\rangle$ that is normal (now with respect to the continued fraction expansion), we show that for any integers $m\ge 2$, $k\ge 1$, the continued fraction $\langle a_k, a_{ m+k},a_{ 2m+k},a_{ 3m+k},\dots\rangle$ will never be normal.
Categories: math.NT math.DS