Packing arithmetic progressions
Noga Alon, Micha{\l} D\k{e}bski, Jaros{\l}aw Grytczuk, and Jakub Przyby{\l}o

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 be a collection of finite arithmetic progressions, where each is an initial segment of the set of consecutive multiples of a positive integer . Let denote the minimum length of an interval containing pairwise disjoint \emph{shifted} copies of all members of the family . We study this parameter in the following two cases: for a fixed positive integer , (1) each progression in has the form , and (2) all progressions of have the same size , that is, . We in particular derive the following asymptotic estimates. In case (1), when , we get . In case (2), when , we get , while if , then…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Analytic Number Theory Research · Mathematical Dynamics and Fractals
