The achievement set of generalized multigeometric sequences

TL;DR

This paper investigates the topological structure of all possible sums of generalized multigeometric series, identifying conditions under which the achievement set forms intervals, Cantor sets, or Cantorvals.

Contribution

It characterizes the topological types of achievement sets for a broad class of multigeometric series, extending understanding of their geometric and topological properties.

Findings

Achievement sets can be intervals, Cantor sets, or Cantorvals.

Specific regions of the parameter space determine the set's topology.

The study provides criteria for the topological classification of these sets.

Abstract

We study the topology of all possible subsums of the generalized multigeometric series $k_1f(x)+k_2f(x)+\dots+k_mf(x)+\dots + k_1f(x^n)+\dots+k_mf(x^n)+\dots,$ where $k_1, k_2, \dots, k_m$ are fixed positive real numbers and $f$ runs along a certain class of non-negative functions on the unit interval. We detect particular regions of this interval for which this achievement set is, respectively, a compact interval, a Cantor set and a Cantorval.