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Density of summable subsequences of a sequence and its applications
Mathematica Slovaca ( IF 0.654 ) Pub Date : 2020-05-23 , DOI: 10.1515/ms-2017-0379
Bingzhe Hou; Yue Xin; Aihua Zhang

Let x = {xn}n=1∞ be a sequence of positive numbers, and 𝓙x be the collection of all subsets A ⊆ ℕ such that ∑k∈Axk < +∞. The aim of this article is to study how large the summable subsequence could be. We define the upper density of summable subsequences of x as the supremum of the upper asymptotic densities over 𝓙x, SUD in brief, and we denote it by D*(x). Similarly, the lower density of summable subsequences of x is defined as the supremum of the lower asymptotic densities over 𝓙x, SLD in brief, and we denote it by D*(x). We study the properties of SUD and SLD, and also give some examples. One of our main results is that the SUD of a non-increasing sequence of positive numbers tending to zero is either 0 or 1. Furthermore, we obtain that for a non-increasing sequence, D*(x) = 1 if and only if lim infk→∞nxn=0, which is an analogue of Cauchy condensation test. In particular, we prove that the SUD of the sequence of the reciprocals of all prime numbers is 1 and its SLD is 0. Moreover, we apply the results in this topic to improve some results for distributionally chaotic linear operators.
更新日期:2020-07-07

 

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