Thinnable Ideals and Invariance of Cluster Points

2017/06/24 by Leonetti, Paolo
#40A35 (Primary) 11B05 #54A20 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.1706.07954

Abstract

We define a class of so-called thinnable ideals I on the positive integers which includes several well-known examples, e.g., the collection of sets with zero asymptotic density, sets with zero logarithmic density, and several summable ideals. Given a sequence (x_n) taking values in a separable metric space and a thinnable ideal I, it is shown that the set of I-cluster points of (x_n) is equal to the set of I-cluster points of almost all its subsequences, in the sense of Lebesgue measure. Lastly, we obtain a characterization of ideal convergence, which improves the main result in [Trans. Amer. Math. Soc. 347 (1995), 1811--1819].

Related

• Characterizations of Ideal Cluster Points
• Limit points of subsequences
• Rough families, cluster points, and cores
• Invariance of Ideal Limit Points
• The Baire Category of Subsequences and Permutations which preserve Limit\n Points