منابع مشابه
Lattice point counting and harmonic analysis
We explain the application of harmonic analysis to count lattice points in large regions. We also present some of our recent results in the three-dimensional case.
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Let d and k be integers with 1 ≤ k ≤ d − 1. Let Λ be a d-dimensional lattice and let K be a d-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of k-dimensional linear subspaces needed to cover all points in Λ ∩ K. In particular, our results imply that the minimum number of k-dimensional linear subspaces needed to cover the d-dimensional n ×...
متن کاملThe Variance of the Hyperbolic Lattice Point Counting Function
The problem of estimating the number of points of a lattice that lie in a ball, is often called the circle problem. In the case of lattices in Euclidean space, this question goes back at least as far as Gauss. If we call Nρ the number of points of Z inside the ball B(0, ρ), then one easily sees that the leading term of Nρ is the area, πρ, of B(0, ρ). It is not difficult to show that the error t...
متن کاملThe number of configurations in lattice point counting I
When a strictly convex plane set S moves by translation, the set J of points of the integer lattice that lie in S changes. The number K of equivalence classes of sets J under lattice translations (configurations) is bounded in terms of the area of the Brunn-Minkowski difference set of S. If S satisfies the Triangle Condition, that no translate of S has three distinct lattice points in the bound...
متن کاملA Simple Algorithm for Lattice Point Counting in Rational Polygons
We propose a simple algorithm for lattice point counting in rational polygons. A rational polygon is one whose vertices have rational coordinates. The algorithm decomposes a given polygon into right trapezoids and counts the number of lattice points in the right trapezoids. Each right trapezoid can be dissected into a rectangle and a right-angled triangle in the obvious way. The number of latti...
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ژورنال
عنوان ژورنال: Foundations of Computational Mathematics
سال: 2015
ISSN: 1615-3375,1615-3383
DOI: 10.1007/s10208-015-9275-7