Sets with Arbitrarily Slow Favard Length Decay

TL;DR

This paper constructs non-self-similar Cantor sets demonstrating that the Favard length of their epsilon-neighborhoods can decay at any arbitrarily slow rate, challenging previous assumptions about decay rates.

Contribution

It introduces a method to create Cantor sets with customizable Favard length decay rates, expanding understanding of geometric measure theory.

Findings

Favard length decay can be arbitrarily slow for certain Cantor sets

Construction of non-self-similar Cantor sets with prescribed decay rates

Challenges existing beliefs about uniform decay behavior

Abstract

In this article, we consider the concept of the decay of the Favard length of $\varepsilon$-neighborhoods of purely unrectifiable sets. We construct non-self-similar Cantor sets for which the Favard length decays arbitrarily with respect to $\varepsilon$.