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Dual vectors and lower bounds for the nearest lattice point problem. (English) Zbl 0653.10026
Let \(L\) be a lattice in \(\mathbb R^ n\) and let \(L^*\) be its dual. The author shows that for each \(x\in\mathbb R^ n\setminus L\) there exists a nonzero \(v\in L^*\) such that \[ \frac{| \{(x,v)\}|}{\| v\|}\geq c_ n\cdot d(x,L), \] where \((x,v)\) is the usual inner product on \(\mathbb R^ n,\) \(\{\alpha\}\) the minimal distance of \(\alpha\) to an integer, \(d(x,L)\) is the distance from \(x\) to \(L\) and \(c_ n\geq (6n^ 2+1)^{-1}.\) The proof is not constructible. The best known constructible proof gives a value \(c_ n\geq 9^{-n}.\)
Reviewer: F. van der Linden

MSC:
11H99 Geometry of numbers
11H31 Lattice packing and covering (number-theoretic aspects)
68R05 Combinatorics in computer science
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