An analogue of Pillai's theorem for continued fraction normality and an application to subsequences

TL;DR

This paper establishes the equivalence of two notions of continued fraction normality, extending Pillai's theorem to continued fractions, and applies this to show that selecting subsequences along arithmetic progressions does not preserve normality.

Contribution

It introduces an analogue of Pillai's theorem for continued fraction normality and uses it to analyze subsequences, providing new insights into continued fraction expansions.

Findings

Proved the equivalence of two notions of continued fraction normality.

Provided a new proof that subsequences along arithmetic progressions do not preserve normality.

Extended Pillai's theorem to the setting of continued fractions.

Abstract

We show that two notions of continued fraction normality, one where overlapping occurrences of finite patterns are counted as distinct occurrences, and another where only disjoint occurrences are counted as distinct, are identical. This equivalence involves an analogue of a theorem due to S. S. Pillai in 1940 for base-$b$ expansions. The proof requires techniques which are fundamentally different, since the continued fraction expansion utilizes a countably infinite alphabet, leading to a non-compact space. Utilizing the equivalence of these two notions, we provide a new proof of Heersink and Vandehey's recent result that selection of subsequences along arithmetic progressions does not preserve continued fraction normality.