Duality between Measure and Category of Almost All Subsequences of a Given Sequence

Let $S$ be the set of subsequences $(x_{n_k})$ of a given real sequence $(x_n)$ which preserve the set of statistical cluster points. It has been recently shown that $S$ is a set of full (Lebesgue) measure. Here, on the other hand, we prove that $S$ is meager if and only if there exists an ordinary limit point of $(x_n)$ which is not a statistical cluster point of $(x_n)$. This provides a non-analogue between measure and category.