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

TL;DR

This paper explores the relationship between measure and category in the context of subsequences of a real sequence, revealing a duality that highlights differences in typical behavior under measure and category.

Contribution

It establishes a novel duality result showing that the set of subsequences preserving statistical cluster points is full measure but meager under certain conditions.

Findings

Set of subsequences with preserved statistical cluster points has full measure.

Such set is meager if and only if an ordinary limit point is not a statistical cluster point.

Reveals a non-analogue between measure and category in subsequence behavior.

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 a statistical cluster point of $(x_n)$. This provides a non-analogue between measure and category.