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

TL;DR

This paper investigates the structure of the set of subsums derived from a convergent series involving sine functions with fixed integer coefficients, revealing that the set can be a union of intervals, a Cantor set, or a Cantorval depending on the parameter.

Contribution

It characterizes the conditions under which the set of subsums forms different types of sets, including intervals, Cantor sets, and Cantorvals, for a series involving trigonometric functions.

Findings

The set of subsums can be a finite union of closed intervals.

The set can also be a Cantor-type set.

Depending on the parameter, the set can be a Cantorval.

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.