Packing Density of Sets With Only Two Nonmixed Gaps

TL;DR

This paper investigates the optimal density of packings of integer sets with only two distinct gap lengths, providing explicit bounds, conjectures on tightness, and proofs for special cases, linking to a Motzkin problem.

Contribution

It introduces a new analysis of packing densities for sets with two gaps, offering explicit bounds and proving cases where one gap length appears only once.

Findings

Derived explicit lower bounds on packing density

Conjectured the bounds are tight, supported by partial proofs

Connected the problem to a Motzkin problem on independence ratios

Abstract

For a finite set of integers such that the first few gaps between its consecutive elements equal $a$, while the remaining gaps equal $b$, we study dense packings of its translates on the line. We obtain an explicit lower bound on the corresponding optimal density, conjecture its tightness, and prove it in case one of the gap lengths, $a$ or $b$, appears only once. This is equivalent to a Motzkin problem on the independence ratio of certain integer distance graphs.