Algebraic and topological properties of some sets in l₁

Abstract

For a sequence x ∈ ℓ1 \ c00, one can consider the set E(x) of all subsums of series ∞ n=1 x(n). Guthrie and Nymann proved that E(x) is one of the following types of sets:(I) a finite union of closed intervals;(C) homeomorphic to the Cantor set;(MC) homeomorphic to the set T of subsums of ∞ n=1 b(n) where b(2n − 1) = 3/4 n and b(2n) = 2/4 n .By I, C and MC denote the sets of all sequences x ∈ ℓ1 \ c00, such that E(x) has the property (I), (C) and (MC), respectively. In this note we show that I and C are strongly c-algebrable and MC is c-lineable. We show that C is a dense G δ -set in ℓ1 and I is a true Fσ-set. Finally we show that I is spaceable while C is not spaceable.

On generating regular Cantorvals connected with geometric Cantor sets

“…Let us observe that Theorem 1 states that 1 can be divided into four disjoint sets c 00 , C, J and MC, where J consists of sequences with achievement sets being finite unions of intervals, C consists of sequences (x n ) with E(x n ) homeomorphic to the Cantor set and MC consists of the sequences with achievement sets homeomorphic to the set T or equivalently to N (MC is the abbrevation of Middle-Cantorval because the structure of the achievement set should be symmetric). Algebraic and topological properties of these subsets of 1 have been recently studied in [2]. All known examples of sequences which achievement sets are Cantorvals belong to the class of multigeometric sequences.…”

Achievement sets on the plane—Perturbations of geometric and multigeometric series

“…Note that Theorem 1.2says that 1 can be divided into four sets: c 00 and the sets with properties prescribed in (1), (2) and (3). Some algebraic and topological properties of these sets have been recently considered in [2].…”

Achievement sets of conditionally convergent series

“…The last result gives the partition of ℓ 1 into four disjoint sets. Topological and algebraic properties of these sets were recently considered in [2,3]. Some sufficient conditions for a given sequence to be a Cantorval were recently described in [1,6,10].…”