Sets of sums of a series depending on sign distributions (Q2322514)

A sequence \((e_{n})\) is called a sign distribution if \(e_{n} =1\) or \(e_{n} =-1\) for every \(n\). A sign distribution \((e_{n})\) is called segmentally alternating if the terms in \((e_{n})\) can be arranged into consecutive finite segments such that, within each segment, there are an equal number of \(e_{n} =1\) and \(e_{n} =-1\). Let \(E\) be the set of all such sign distributions. The author considers also monotone decreasing sequences \((a_{n})\) with limit zero for which \(\sum_{n=1}^{\infty }a_{n} \) diverges and defines the set \(S(E,a_{n})=\left\{\sum_{n=1}^{\infty }e_{n} {\kern 1pt} a_{n} :(e_{n})\in E \right\}\). The purpose of the paper is to study topological properties of such sets. Several interesting results are proved. For example, it is shown that, depending on the sequence \((a_{n})\), the set \(S(E,a_{n})\) can be a proper subset of the real line (but dense in it) or coincide with it.