On the Packing Functions of some Linear Sets of Lebesgue Measure Zero

TL;DR

This paper investigates the relationship between Minkowski measurability and packing asymptotics for certain fractal sets on the real line, providing explicit constants and exploring the limits as the dimension approaches one.

Contribution

It establishes Minkowski measurability as a sufficient condition for packing asymptotics and derives an explicit proportionality constant depending only on the Minkowski dimension.

Findings

Minkowski measurability guarantees the existence of packing asymptotics.

Derived an explicit constant p_d relating packing limits and Minkowski content.

Showed the limit of p_d as d approaches 1 using the Digamma function.

Abstract

We use a characterization of Minkowski measurability to study the asymptotics of best packing on cut-out subsets of the real line with Minkowski dimension $d\in(0,1)$. Our main result is a proof that Minkowski measurability is a sufficient condition for the existence of best packing asymptotics on monotone rearrangements of these sets. For each such set, the main result provides an explicit constant of proportionality $p_d,$ depending only on the Minkowski dimension $d,$ that relates its packing limit and Minkowski content. We later use the Digamma function to study the limiting value of $p_d$ as $d\to 1^-.$ For sharpness, we use renewal theory to prove that the packing constant of the $(1/2,1/3)$ Cantor set is less than the product of its Minkowski content and $p_d$. We also show that the measurability hypothesis of the main theorem is necessary by demonstrating that a monotone rearrangement of the complementary intervals of the 1/3 Cantor set has Minkowski dimension $d=\log2/\log3\in(0,1),$ is not Minkowski measurable, and does not have convergent first-order packing asymptotics. The aforementioned characterization of Minkowski measurability further motivates the asymptotic study of an infinite multiple subset sum problem.