منابع مشابه
Lattice Points on Ellipses
Given a square free positive integer d one may consider the arithmetical function rd(n) = #{n = x + dy/x, y ∈ Z} which can also be described as the number of lattice points on the ellipse x + dy = n and it has a natural interpretation inside the ring of algebraic integers of the field Q( √−d). The main purpose of this paper is to analyse closely this function in connection with the distribution...
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We prove that the lattice points on the circles x2 + y2 = n are well distributed for most circles containing lattice points.
متن کاملClose Lattice Points on Circles
We classify the sets of four lattice points that all lie on a short arc of a circle which has its center at the origin; specifically on arcs of length tR on a circle of radius R, for any given t > 0. In particular we prove that any arc of length ( 40 + 40 3 √ 10 )1/3 R on a circle of radius R, with R > √ 65, contains at most three lattice points, whereas we give an explicit infinite family of 4...
متن کاملNotes on lattice points of zonotopes and lattice-face polytopes
Minkowski’s second theorem on successive minima gives an upper bound on the volume of a convex body in terms of its successive minima. We study the problem to generalize Minkowski’s bound by replacing the volume by the lattice point enumerator of a convex body. To this we are interested in bounds on the coefficients of Ehrhart polynomials of lattice polytopes via the successive minima. Our resu...
متن کاملLattice Points inside Lattice Polytopes
We show that, for any lattice polytope P ⊂ R, the set int(P ) ∩ lZ (provided it is non-empty) contains a point whose coefficient of asymmetry 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 9 · (8l+ 7) d+1 . This implies that the maximum volume of a lattice polytope P ⊂ R d containing exactly k ≥ 1 points of lZ in its interior, is...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1969
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-15-2-199-203