Fundamental rectangles of admissible lattices

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<...<v_k<... with , such that for u>μ the rectangle P(u, v_k)+X contains T=S(P)+R points of Λ, where |R|<5; here S(P) is the area of the rectangle. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 82–89. Translated by O. A. Ivanov.