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

In the study of normal numbers using the base-b representation, b > 2, we know that a number is normal if and only if it is simply normal in all the bases bn, n > 1. We prove the analogue of this result for continued fraction normality. In particular, 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 Pillai (Proc. Indian Acad. Sci. Sect. A 11 (1940) 73-80) 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 (Heersink and Vandehey, Arch. Math. 106 (2016) 363-370).