The achievement set of generalized multigeometric sequences

We study the topology of all possible subsums of the generalized multigeometric series $k_{1}f(x)+k_{2}f(x)+\cdots +k_{m}f(x)+\cdots +k_{1}f(x^n)+\cdots +k_{m}f(x^n)+\dots ,$ 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.