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

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