Packing arithmetic progressions

TL;DR

This paper investigates the minimal interval length needed to pack shifted copies of finite arithmetic progressions, providing asymptotic estimates for different configurations and sizes of these progressions.

Contribution

It introduces new asymptotic bounds for packing arithmetic progressions with shifted copies, extending understanding of their combinatorial and number-theoretic properties.

Findings

For fixed n, when k=n, m(F) = Θ(n^{3/2}/ln n).

For fixed n, when k=n, m(F) = Θ(n^3/ln n).

If k exceeds a certain threshold, m(F) < 3kn.

Abstract

Let $\mathcal{F}=\{A_1,A_2,\ldots,A_k\}$ be a collection of finite arithmetic progressions, where each $A_d$ is an initial segment of the set $D_d=\{d,2d,3d,\ldots\}$ of consecutive multiples of a positive integer $d$. Let $m(\mathcal{F})$ denote the minimum length of an interval containing pairwise disjoint \emph{shifted} copies of all members of the family $\mathcal{F}$. We study this parameter in the following two cases: for a fixed positive integer $n$, (1) each progression in $\mathcal{F}$ has the form $A_d=D_d\cap\{1,2,\ldots,n\}$, and (2) all progressions $A_d$ of $\mathcal{F}$ have the same size $n$, that is, $A_d=D_d\cap \{1,2,\ldots, nd\}$. We in particular derive the following asymptotic estimates. In case (1), when $k=n$, we get $m(\mathcal{F})=\Theta(n^{3/2}/\ln n)$. In case (2), when $k=n$, we get $m(\mathcal{F})=\Theta(n^3/\ln n)$, while if $k>k_0(n)$, then $m(\mathcal{F}) < 3kn$. In both cases we additionally determine $m(\mathcal{F})$ asymptotically or settle its order of magnitude for all $k<n$.