Journal of Mathematical Sciences

, Volume 79, Issue 5, pp 1320–1324 | Cite as

Fundamental rectangles of admissible lattices

  • Kh. Kh. Ruzimuradov
Article
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Abstract

Let Λ be a unimodular lattice in ℝ2, μ a homogeneous minimum of Λ; let P(a,b)⊂ℝ2 be a rectangle with vertices at the points (a,0), ...(0,b), P(a, b)+X its image under the translation by a vector X ∈ ℝ2. We prove that there exists a sequence of positive numbers v1<v2<...<vk<... with\(2\sqrt {2\mu } ^{ - 2} \upsilon _{k - 1} > \upsilon _k\), such that for u>μ the rectangle P(u, vk)+X contains T=S(P)+R points of Λ, where |R|<5; here S(P) is the area of the rectangle. Bibliography: 4 titles.

Keywords

Unimodular Lattice Admissible Lattice Homogeneous Minimum Fundamental Rectangle 

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Literature Cited

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    J. W. S. Cassels,An Introduction to the Geometry of Numbers, Springer, Berlin et al. (1959).Google Scholar
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    B. F. Skubenko, “Minimums of decomposable forms of degreen inn variables,”Zap. Nauchn. Semin. LOMI,183, 142–154 (1990).MATHGoogle Scholar
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    M. M. Skriganov, “Uniform distributions and geometry of numbers” [in Russian], Preprint LOMI P-6-91, Leningrad (1991).Google Scholar
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    M. M. Skriganov, “On integer points in polygons,”Prépublication de l'Institut Fourier,204 (1992).Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Kh. Kh. Ruzimuradov

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