scholarly journals Lattice points in large convex bodies, II

1992 ◽  
Vol 62 (3) ◽  
pp. 285-295 ◽  
Author(s):  
Ekkehard Krätzel ◽  
Werner Nowak
Keyword(s):  
Convex Bodies ◽  
Lattice Points

Asymptotic Formulae for the Lattice Point Enumerator

1999 ◽  
Vol 51 (2) ◽  
pp. 225-249 ◽  
Author(s):  
U. Betke ◽  
K. Böröczky
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Positive Curvature ◽  
Convex Bodies ◽  
Mixed Volume ◽  
Lattice Points ◽  
Asymptotic Formulae ◽  
General Convex ◽  
Lattice Surface ◽  
Small Deformations

AbstractLet M be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hlawka for large λ the number of lattice points in λM is given by G(λM) = V(λM) + O(λd−1−ε(d)) for some positive ε(d). Here we give for general convex bodies the weaker estimatewhere SZd (M) denotes the lattice surface area of M. The term SZd is optimal for all convex bodies and o(λd−1) cannot be improved in general. We prove that the same estimate even holds if we allow small deformations of M.Further we deal with families {Pλ} of convex bodies where the only condition is that the inradius tends to infinity. Here we havewhere the convex body K satisfies some simple condition, V(Pλ; K; 1) is some mixed volume and S(Pλ) is the surface area of Pλ.

Stable Lattices

1953 ◽  
Vol 5 ◽  
pp. 261-270 ◽  
Author(s):  
Harvey Cohn
Keyword(s):  
Convex Body ◽  
Finite Number ◽  
Quadratic Forms ◽  
Convex Bodies ◽  
Lattice Points ◽  
Three Dimensions ◽  
General Convex ◽  
Critical Lattice ◽  
The Absolute

The consideration of relative extrema to correspond to the absolute extremum which is the critical lattice has been going on for some time. As far back as 1873, Korkine and Zolotareff [6] worked with the ellipsoid in hyperspace (i.e., with quadratic forms), and later Minkowski [8] worked with a general convex body in two or three dimensions. They showed how to find critical lattices by selection from among a finite number of relative extrema. They were aided by the long-recognized premise that only a finite number of lattice points can enter into consideration [1] when one deals with lattices “admissible to convex bodies.”

Convex Bodies and Lattice Points

1975 ◽  
Vol 48 (2) ◽  
pp. 110 ◽  
Author(s):  
P. R. Scott
Keyword(s):  
Convex Bodies ◽  
Lattice Points

scholarly journals Mean square discrepancy bounds for the number of lattice points in large convex bodies

2002 ◽  
Vol 87 (1) ◽  
pp. 209-230 ◽  
Author(s):  
Alexander Iosevich ◽  
Eric Sawyer ◽  
Andreas Seeger
Keyword(s):  
Convex Bodies ◽  
Lattice Points ◽  
Mean Square

scholarly journals On lattice points in large convex bodies

2012 ◽  
Vol 151 (1) ◽  
pp. 83-108 ◽  
Author(s):  
Jingwei Guo
Keyword(s):  
Convex Bodies ◽  
Lattice Points

scholarly journals On convex bodies containing a given number of lattice points

1977 ◽  
Vol 9 (2) ◽  
pp. 240-246 ◽  
Author(s):  
G.D. Chakerian ◽  
H. Groemer
Keyword(s):  
Convex Bodies ◽  
Lattice Points
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