Fundamental rectangles of admissible lattices

Kh. Kh. Ruzimuradov

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 11, Tome 204 (1993), pp. 82-89

Résumé

Let $\Lambda$ be a unimodular lattice in $\mathbb R^2$, $\mu$ a homogeneous minimum of $\Lambda$; let $P(a,b)\subset\mathbb R^2$ be a rectangle with vertices at the points $(a,0),\dots,(0,b)$, $P(a,b)+X$ its image under the translation by a vector $X\in\mathbb R^2$. We prove that there exists a sequence of positive numbers $v_1<v_2<\dots<v_k<\dots$ with $2\sqrt2\mu^{-2}v_{k-1}>v_k$, such that for $u>\mu$ the rectangle $P(u,v_k)+X$ contains $T=S(P)+R$ points of $\Lambda$, where $|R|<5$; here $S(P)$ is the area of the rectangle. Bibliography: 4 titles.