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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
##### Keywords:
dual lattice; lattice basis; lattice; homogeneous minimum
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##### References:
 [1] L. Babai, On Lovász’ lattice reduction and the nearest lattice point problem,Combinatorica,6 (1986), 1–13. · Zbl 0593.68030 · doi:10.1007/BF02579403 [2] J. W. S. Cassels,An Introduction to the Geometry of Numbers, Springer Verlag, Heidelberg 1971. · Zbl 0209.34401 [3] P.Van Emde Boas, AnotherNP-complete problem and the complexity of computing short vectors in a lattice,Math. Dept. Report 81-04. Univ. of Amsterdam, April 1981. [4] R.Kannan, Minkowski’s convex body theorem and integer programming,to appear in Mathematics of Operations Research. · Zbl 0639.90069 [5] A. I.Khinchin, A quantitative formulation of Kronecker’s theory of approximation,Inv. Akad. Nauk. SSSR (ser. Mat),12 113–122 (in russian). · Zbl 0030.02002 [6] A. Korkine andG. Zolotarev, Sur les formes quadratiques,Mathematiche Annalen,6 (1973), 366–389. · JFM 05.0109.01 · doi:10.1007/BF01442795 [7] J. C.Lagarias, H. W.Lenstra and C. P.Schnorr, Korkine-Zolotarev bases and the successive minima of a lattice and its reciprocal lattice,to appear in Combinatorica. · Zbl 0723.11029 [8] A. K. Lenstra, H. W. Lenstra andL. Lovász, Factoring polynomials withrational coefficients,Math. Ann. 261 (1982), 515–534. · Zbl 0488.12001 · doi:10.1007/BF01457454 [9] L.Lovász,personal communication.
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