On the packing functions of some linear sets of Lebesgue measure zero

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 $d\to 1^{-}$. This study further provides two sharpness results illustrating the necessity of the hypotheses of the main result. Finally, the aforementioned characterization of Minkowski measurability motivates the asymptotic study of an infinite multiple subset sum problem.