Cantorvals as sets of subsums for a series connected with trigonometric functions

Abstract

We study properties of the set of subsums for a convergent series $ k_1 \sin x + \dots + k_m \sin x +\dots + k_1\sin x^n +\dots + k_m \sin x^n + \dots $, where $k_1, k_2, k_3,\dots,k_m$ are fixed positive integers and $0<x<1$. Depends on parameter $x$ this set can be a finite union of closed intervals or Cantor-type set or even Cantorval.