Abstract
New estimates for the defect of the admissible set in a lattice are obtained for a sufficiently large class of sequences.
Similar content being viewed by others
References
A. M. Raigorodskii, “The defects of admissible balls and octahedra in a lattice, and systems of generic representatives,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 189, No. 6, 117-141 (1998).
N. G. Moshchevitin, “The defect of an admissible octahedron in a lattice,” Mat. Zametki [Math. Notes], 58, 558-568 (1995).
T. W. Cusick and J. Wolfskill, “Lattice octahedra and sums of powers of linear forms,” J. London Math. Soc. (2), 38, 207-215 (1988).
L. J. Mordell, “Lattice octahedra,” Canad. J. Math., 12, 297-302 (1960).
R. Bantegnie, “Le problème des octaèdres en dimension 5,” Acta Arith., 14, 185-202 (1968).
U. Wessels, Die Sätze von White und Mordellüber kritische Gitter von Polytopen in den Dimensionen 4 und 5, Diplomarbeit, Mathematisches Institut der Ruhr-Universität Bochum (1989).
S. S. Ryshkov, “On the problem of the determination of the perfect quadratic forms in many variables,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 142, 215-239 (1976).
N. V. Zakharova, “Centerings of 8-dimensional lattices that preserve a frame of successive minima,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 152, 97-123 (1980).
A. M. Raigorodskii, “The defects of admissible sets in a lattice, and systems of common representatives,” Grazer Mathematische Berichte, No. 338, 31-62 (1999).
P. Erdös and J. Spencer, Probabilistic Methods in Combinatorics, Akadémiai Kiadó, Budapest (1974).
N. N. Kuzyurin, “Asymptotic investigation of the problem of covering,” Problemy Kibernetiki, No. 37, 19-56 (1980).
A. M. Raigorodskii, “Systems of common representatives,” Fundament. i Prikl. Mat., 5, No. 3, 851-860 (1999).
V. E. Tarakanov, Combinatorial Problems and (0, 1) –Matrices [in Russian], Fizmatlit, Moscow (1985).
P. M. Gruber and C. G. Lekkerkerker, Geometry of Numbers, North-Holland, Amsterdam (1987).
J. W. S. Cassels, An Introduction to the Geometry of Numbers, Springer-Verlag, Berlin-Göttingen-Heidelberg (1959).
N. M. Korobov, Number-Theoretic Methods in Approximate Analysis [in Russian], Fizmatlit, Moscow (1963).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Raigorodskii, A.M. A Probabilistic Approach to the Problem of the Defects of Admissible Sets in a Lattice. Mathematical Notes 68, 770–774 (2000). https://doi.org/10.1023/A:1026616918102
Issue Date:
DOI: https://doi.org/10.1023/A:1026616918102