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

Abstract

We study properties of the set of subsums for convergent series k1 sin x + ... + km sin x + ... + k1 sin x[(n-1)/m+1] + ... + km sin x[(n-1)/m+1] + ... where k1, k2, k3, ..., km are fixed positive integers and 0<x<1. It is proved that depending on the parameter x this set can be a finite union of closed intervals or Cantor-type set or even Cantorval.