Fundamental rectangles of admissible lattices
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 11, Tome 204 (1993), pp. 82-89
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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 $v_1$ 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.
@article{ZNSL_1993_204_a5,
author = {Kh. Kh. Ruzimuradov},
title = {Fundamental rectangles of admissible lattices},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {82--89},
publisher = {mathdoc},
volume = {204},
year = {1993},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_1993_204_a5/}
}
Kh. Kh. Ruzimuradov. Fundamental rectangles of admissible lattices. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 11, Tome 204 (1993), pp. 82-89. http://geodesic.mathdoc.fr/item/ZNSL_1993_204_a5/