Geodesic
Upper bound for the product of nonhomogeneous linear forms
Matematičeskie zametki, Tome 23 (1978) no. 6, pp. 789-797

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It is proved that for any unimodular lattice $\Lambda$ with homogeneous minimum $L>0$ and any set of real numbers $\alpha_1,\alpha_2,\dots,\alpha_n$ there exists a point ($y_1, y_2,\dots,y_n$) of $\Lambda$ such that $$ \prod_{1\le i\le n}|y_i+\alpha_i|\le2^{-n/2_\gamma n}(1+3L^{8/(3^n)/(\gamma^{2/3}-2L^{8/(3^n)})})^{-n/2}, $$ where $\gamma^n=n^{n/(n-1)}$. 
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     author = {K. Bakiev and A. S. Pen and B. F. Skubenko},
     title = {Upper bound for the product of nonhomogeneous linear forms},
     journal = {Matemati\v{c}eskie zametki},
     pages = {789--797},
     publisher = {mathdoc},
     volume = {23},
     number = {6},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1978_23_6_a1/}
}
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K. Bakiev; A. S. Pen; B. F. Skubenko. Upper bound for the product of nonhomogeneous linear forms. Matematičeskie zametki, Tome 23 (1978) no. 6, pp. 789-797. http://geodesic.mathdoc.fr/item/MZM_1978_23_6_a1/