Multiple Lattice Packings and Coverings of the Plane with Triangles

2014/12/16 by Kirati Sriamorn, Sriamorn, Kirati
Materials Science · Mathematics · #Analytic Number Theory Research #Analytic and geometric function theory #FOS: Mathematics #Metric Geometry (math.MG) #Number Theory (math.NT) #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.1412.5096

openalex publication_date 2014/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a convex disk K and a positive integer j, let δ_L^j(K) and ϑ_L^j(K) denote the j-fold lattice packing density and the j-fold lattice covering density of K, respectively. I will prove that for every triangle T we have that δ_L^j(T)=(2j²)/(2j+1) and ϑ_L^j(T)=(2j+1)/(2). Furthermore, I also obtain that the numbers of lattices which attain these densities both are (2j+1)∏_p|2j+1(1-(2)/(p)), where the product is over the distinct prime numbers dividing 2j+1.

Related

• On the Lattice Packings and Coverings of the Plane with Convex Quadrilaterals
• The set of packing and covering densities of convex disks
• On densities of lattice arrangements intersecting every i-dimensional\n affine subspace
• Dense packing of space with various convex solids
• Multiple arrangements