Macdonald's solid-angle sum for real dilations of rational polygons

2016/02/08 by Quang-Nhat Le, Le, Quang-Nhat, Sinai Robins +1
Computer Science · Mathematics · #32A27 #52C10 #52C15 #52C17 #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Metric Geometry (math.MG) #math.CO #math.MG #msc:32A27 #msc:52C10 #msc:52C15 #msc:52C17

paper · pdf · doi:10.48550/arxiv.1602.02681

20 pages, 4 figures

arxiv created 2016/02/08 · openalex publication_date 2016/02/08 · arxiv updated 2016/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The solid-angle sum A_P (t) of a rational polytope P ⊂ ℝ^d, with t ∈ ℤ was first investigated by I.G. Macdonald. Using our Fourier-analytic methods, we are able to establish an explicit formula for A_P (t), for any real dilation t and any rational polygon P ⊂ ℝ². Our formulation sheds additional light on previous results, for lattice-point enumerating functions of triangles, which are usually confined to the case of integer dilations. Our approach differs from that of Hardy and Littlewood in 1992, but offers an alternate point of view for enumerating weighted lattice points in real dilations of real triangles.

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