Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Minimum packing density for sets of four integers

Cindy Li, David Offner, Iris Ye

TL;DR

The paper determines the minimum packing density for sets of four integers under the $S$-packing framework. It establishes a universal lower bound $d_p(S) \ge \tfrac{1}{|\mathrm{diff}(S)|}$ via a greedy packing construction, and a corresponding upper bound when $\mathrm{diff}(S)$ contains a consecutive block, yielding $d_p(S) \le \tfrac{1}{n+1}$ for $\{0,\dots,n\} \subseteq \mathrm{diff}(S)$. Combining these results shows that for any $|S|=4$, $d_p(S) \ge \tfrac{1}{7}$, and that $S=\{0,1,4,6\}$ attains equality with $d_p(S)=\tfrac{1}{7}$. The constructive greedy approach also demonstrates that the densest packing in the four-element case is realized by a periodic set, specifically $A=\{7n\}$. This identifies the exact minimum packing density among all four-element integer sets and provides a concrete extremal example with practical implications for related packing and covering problems in number theory.

Abstract

We prove that the set $\{0, 1, 4, 6\}$ achieves the minimum packing density among all sets of integers with cardinality four, with a density of $\frac{1}{7}$.

Minimum packing density for sets of four integers

TL;DR

The paper determines the minimum packing density for sets of four integers under the -packing framework. It establishes a universal lower bound via a greedy packing construction, and a corresponding upper bound when contains a consecutive block, yielding for . Combining these results shows that for any , , and that attains equality with . The constructive greedy approach also demonstrates that the densest packing in the four-element case is realized by a periodic set, specifically . This identifies the exact minimum packing density among all four-element integer sets and provides a concrete extremal example with practical implications for related packing and covering problems in number theory.

Abstract

We prove that the set achieves the minimum packing density among all sets of integers with cardinality four, with a density of .
Paper Structure (3 sections, 9 theorems, 32 equations)

This paper contains 3 sections, 9 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. An upper bound on packing density
  3. A lower bound on packing density

Key Result

Proposition 1

Suppose $S, A \subseteq \mathbb{Z}$. Then $A$ is $S$-packing if and only if for all $a,b \in A$ with $a<b$, $b-a \notin \mathrm{diff}(S)$.

Theorems & Definitions (9)

  • Proposition 1
  • Proposition 2
  • Corollary 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Proposition 7
  • Corollary 8
  • Corollary 9