Abstract
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 also a statistical cluster point of \((x_n)\). This provides a non-analogue between measure and category.
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Leonetti, P., Miller, H.I. & Wieren, L.MV. Duality between measure and category of almost all subsequences of a given sequence. Period Math Hung 78, 152–156 (2019). https://doi.org/10.1007/s10998-018-0255-y
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DOI: https://doi.org/10.1007/s10998-018-0255-y