Average number of lattice points in a disk
نویسندگان
چکیده
منابع مشابه
The Number of Lattice Points in a ¿ - dimensional Hypersphere
Thus for fc ̂ 4 8k = fc/2 1. The value of fc which has received the greatest attention is fc = 2, the number of lattice points in a circle. Wilton [2] gives an account of the early work in this problem. Since that time several results have been published establishing new values of 8 for which P2(x) = 0(xe). One of the most recent is Chen Jing-ren's proof [3] that P2(x) = 0(x12/37). Hardy (see [2...
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The parametric lattice-point counting problem is as follows: Given an integer matrix A ∈ Zm×n , compute an explicit formula parameterized by b ∈ R that determines the number of integer points in the polyhedron {x ∈R : Ax É b}. In the last decade, this counting problem has received considerable attention in the literature. Several variants of Barvinok’s algorithm have been shown to solve this pr...
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Let L be a lattice in R. We upper bound the number of points of L contained in a small sphere, centered anywhere in R. One way to do this is based on the observation that if the radius of the sphere is sufficiently small then the lattice points contained in that sphere give rise to a spherical code with a certain minimum angle. Another method involves Gaussian measures on L in the sense of [2]....
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We give bijective proofs that, when combined with one of the combinatorial proofs of the general ballot formula, constitute a combinatorial argument yielding the number of lattice paths from (0, 0) to (n, rn) that touch or cross the diagonal y = rx at exactly k lattice points. This enumeration partitions all lattice paths from (0, 0) to (n, rn). While the resulting formula can be derived using ...
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We show that, for any lattice polytope P ⊂ R, the set int(P ) ∩lZ (provided it is non-empty) contains a point whose coefficient ofasymmetry with respect to P is at most 8d · (8l+7)2d+1. If, moreover,P is a simplex, then this bound can be improved to 8 · (8l+ 7)d+1.As an application, we deduce new upper bounds on the volume ofa lattice polytope, given its ...
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ژورنال
عنوان ژورنال: Communications on Pure and Applied Analysis
سال: 2015
ISSN: 1534-0392
DOI: 10.3934/cpaa.2016.15.1