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W4367694676 abstract "For $dinmathbb{N}$, a compact sphere packing of Euclidean space $mathbb{R}^{d}$ is a set of spheres in $mathbb{R}^{d}$ with disjoint interiors so that the contact hypergraph of the packing is the vertex scheme of a homogeneous simplicial $d$-complex that covers all of $mathbb{R}^{d}$. We are motivated by the question: For $d,ninmathbb{N}$ with $d,ngeq2$, how many configurations of numbers $0<r_{0}<r_{1}<ldots<r_{n-1}=1$ can occur as the radii of spheres in a compact sphere packing of $mathbb{R}^{d}$ wherein there occur exactly $n$ sizes of sphere? We introduce what we call `heteroperturbative sets' of labeled triangulations of unit spheres and we discuss the existence of non-trivial examples of heteroperturbative sets. For a fixed heteroperturbative set, we discuss how a compact sphere packing may be associated to the heteroperturbative set or not. We proceed to show, for $d,ninmathbb{N}$ with $d,ngeq2$ and for a fixed heteroperturbative set, that the collection of all configurations of $n$ distinct positive numbers that can occur as the radii of spheres in a compact packing is finite, when taken over all compact sphere packings of $mathbb{R}^{d}$ which have exactly $n$ sizes of sphere and which are associated to the fixed heteroperturbative set." @default.
W4367694676 created "2023-05-03" @default.
W4367694676 creator A5014433071 @default.
W4367694676 creator A5071335946 @default.
W4367694676 date "2023-05-01" @default.
W4367694676 modified "2023-10-18" @default.
W4367694676 title "On compact packings of Euclidean space with spheres of finitely many sizes" @default.
W4367694676 doi "https://doi.org/10.48550/arxiv.2305.00758" @default.
W4367694676 hasPublicationYear "2023" @default.
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