منابع مشابه
Lattice 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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This article is concerned with the number B D (x) of integer points with relative prime coordinates in p x D, where x is a large real variable and D is a starlike set in the Euclidean plane. Assuming the truth of the Riemann Hypothesis, we establish an asymptotic formula for B D (x). Applications to certain special geometric and arithmetic problems are discussed .
متن کاملOn the Distribution of Primitive Lattice Points in the Plane
This is one form of Kronecker’s theorem [4-j aura, since N can be chosen arbitrarily large, it follows that there are an infinity of integer sets (x, yl, -. . , y,) with x > 0 satisfying (A). For n > 2, it is not possible to strengthen this result by replacing the & in (A), throughout, by any function y/(x) which tends to zero as x-m (see, e.g., 15) , Kap VII,37, Satz 6). But, in the case n = 1...
متن کاملConvex Lattice Polytopes and Cones with Few Lattice Points Inside, from a Birational Geometry Viewpoint
It is pretty well-known that toric Fano varieties of dimension k with terminal singularities correspond to convex lattice polytopes P ⊂ R of positive finite volume, such that P ⋂ Z consists of the point 0 and vertices of P (cf., e.g. [10], [36]). Likewise, Q−factorial terminal toric singularities essentially correspond to lattice simplexes with no lattice points inside or on the boundary (excep...
متن کاملLattice Points in 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 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: Czechoslovak Mathematical Journal
سال: 2005
ISSN: 0011-4642,1572-9141
DOI: 10.1007/s10587-005-0043-8