Continued fraction normality is not preserved along arithmetic progressions

TL;DR

This paper demonstrates that unlike base-$b$ expansions, continued fraction normality is not preserved along arithmetic progressions, showing a fundamental difference in the behavior of these two types of expansions.

Contribution

It proves that continued fraction normality is not maintained under arithmetic subsequences, contrasting with the known preservation in base-$b$ expansions.

Findings

Continued fraction normality is not preserved along arithmetic progressions.

Base-$b$ expansion normality is preserved under such progressions.

The result highlights a fundamental difference between base-$b$ and continued fraction normality.

Abstract

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.