Continued fraction normality is not preserved along arithmetic progressions

Byron Heersink; Joseph Vandehey

paper

Continued fraction normality is not preserved along arithmetic progressions

arXiv:1509.05501

Abstract

It is well known that if $0. a_{1} a_{2} a_{3} \dots$ is the base- $b$ expansion of a number normal to base- $b$ , then the numbers $0. a_{k} a_{m + k} a_{2 m + k} \dots$ for $m \geq 2$ , $k \geq 1$ are all normal to base- $b$ as well. In contrast, given a continued fraction expansion $⟨ a_{1}, a_{2}, a_{3}, \dots ⟩$ that is normal (now with respect to the continued fraction expansion), we show that for any integers $m \geq 2$ , $k \geq 1$ , the continued fraction $⟨ a_{k}, a_{m + k}, a_{2 m + k}, a_{3 m + k}, \dots ⟩$ will never be normal.