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The Lebesgue measure of boundaries of multigeometric Cantorvals

Piotr Nowakowski, Franciszek Prus-Wiśniowski

TL;DR

The paper addresses the question of boundary regularity for achievement sets $E(x_n)$, focusing on multigeometric Cantorvals. It proves that the boundary of any multigeometric Cantorval has Lebesgue measure $0$ and extends this to a broad class of standard Cantorvals using self-similarity and Lebesgue density arguments, with Kyiv Cantorvals shown to be standard as well. Additionally, it analyzes the set of unique representations $U(x_n)$, establishing $U$ as a $G_ u$ set and providing criteria under which $E(x_n)$ is Cantor in the semi-fast setting, while highlighting open questions about subsequences producing Cantorvals. Together, these results deepen the understanding of the fine structure and topological types of achievement sets, including both multigeometric and non-multigeometric examples, and clarify the role of uniqueness in their representations.

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 $\mathcal{G}_δ$.

The Lebesgue measure of boundaries of multigeometric Cantorvals

TL;DR

The paper addresses the question of boundary regularity for achievement sets , focusing on multigeometric Cantorvals. It proves that the boundary of any multigeometric Cantorval has Lebesgue measure and extends this to a broad class of standard Cantorvals using self-similarity and Lebesgue density arguments, with Kyiv Cantorvals shown to be standard as well. Additionally, it analyzes the set of unique representations , establishing as a set and providing criteria under which is Cantor in the semi-fast setting, while highlighting open questions about subsequences producing Cantorvals. Together, these results deepen the understanding of the fine structure and topological types of achievement sets, including both multigeometric and non-multigeometric examples, and clarify the role of uniqueness in their representations.

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 .
Paper Structure (4 sections, 18 theorems, 71 equations)

This paper contains 4 sections, 18 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. The Lebesgue measure of the boundary
  3. The Kyiv Cantorvals
  4. Remarks on the set of uniqueness

Key Result

Theorem 1

$E(x_n)$ is a multi-interval set if and only if $K(x_n)$ is finite.

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • Lemma 7
  • ...and 23 more