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

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.