The Lebesgue measure of boundaries of multigeometric Cantorvals

TL;DR

This paper proves that the boundaries of multigeometric Cantorvals have Lebesgue measure zero and explores their set of uniqueness, showing it always belongs to a specific Borel class, thus advancing understanding of their measure and topological properties.

Contribution

It establishes that boundaries of multigeometric Cantorvals are null sets and characterizes their sets of uniqueness within a Borel class, extending previous results to a broader class.

Findings

Boundaries of multigeometric Cantorvals have Lebesgue measure zero.

Sets of uniqueness of achievement sets are always Borel $ extstyle ext{G}_ ext{delta}$ sets.

Extended measure-zero boundary result to a larger class of Cantorvals.

Abstract

We prove that the boundary of every multigeometric Cantorval is a null set, and extend this result to a larger class of standard achievable Cantorvals. In addition, we discuss the sets of uniqueness of achievement sets and show that they always belong to the Borel class $\mathcal{G}_\delta$.