Achievable Cantorvals almost without reversed Kakeya conditions

TL;DR

This paper constructs examples of achievable Cantorvals with minimal reversed Kakeya conditions and zero boundary measure, advancing understanding of their geometric properties and addressing specific open problems.

Contribution

It introduces a method to construct achievable Cantorvals with reversed Kakeya conditions on a zero-density set, and shows their boundaries have zero Lebesgue measure.

Findings

Achievable Cantorvals can be constructed with reversed Kakeya conditions on a zero-density set.

The boundaries of these Cantorvals have Lebesgue measure zero.

The paper does not resolve whether achievable Cantorvals with positive boundary measure exist.

Abstract

Examples of achievable Cantorvals are constructed with reversed Kakeya conditions only on a set of asymptotic density zero which answers in positive the Problem 5.2 from Marchwicki and Miska (2021). Additionally, the Lebesgue measure of the boundaries of these Cantorvals is found to be zero which does not answer the still open problem of existence of achievable Cantorvals with boundaries of positive measure.