the center of distances for some multigeometric series

Abstract

Basic properties of the center of distances of a set are investigated. Computation of the center for achievement sets is particularly aimed at. A new sufficient condition for the center of distances of the set of subsums of a fast convergent series to consist of only 0 and the terms of the series is found. A complete description of the center of distances for some particular type of multigeometric series is provided. In particular, the center of distances C(E) is the union of all centers of distances of C(Fn) where Fn is the set of all ninitial subsums of the discussed multigeometric series satisfying some special requirements. Several open questions are raised.

On the $c_0$-equivalence and permutations of series

“…, what, defining σ(j) := σ 1 (j) for k n + 1 ≤ j ≤ k n+1 gives us (3). Of course we can assume that k n+1 is greater than k ′ n+1 .…”

“…Plewik and M. Walczyńska [7, Theorem 2.1] have generalized one implication of the von-Neumann's theorem using the so-called "back-and-forth" method. In a paper by M. Banakiewicz, A. Bartoszewicz and F. Prus-Wiśniowski [3] the reverse implication to this theorem was proved. Given a metric space (X, d) the center of distances of X is defined as Z(X) = {α ≥ 0 : ∀x ∈ X, ∃y ∈ X such that d(x, y) = α}.…”