The Achievement Set of Generalized Multigeometric Sequences

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.