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A probabilistic approach to the problem of the defects of admissible sets in a lattice. (English. Russian original) Zbl 1015.11029
Math. Notes 68, No. 6, 770-774 (2000); translation from Mat. Zametki 68, No. 6, 910-916 (2000).
Let $$L$$ be a nondegenerate homogeneous lattice in $$\mathbb R^n$$ containing the integer lattice $$\mathbb Z^n$$ as a sublattice. The defect $$d(L)$$ of $$E$$ in $$L$$ is defined as the smallest number of vectors to be deleted from the frame $$E=\{o,e_1,\dots,e_n\}$$ made up of the origin and the unit coordinate vectors such that the remaining system be complementable to a basis of $$L$$. Consider an arbitrary sequence of sets $$\Omega_1,\Omega_2,\dots$$ such that, for each $$n\in \mathbb N$$, $$\Omega_n$$ contains $$\pm E$$ and is contained in the cube $$[-1,1]^n$$; moreover, the intersection of $$\Omega_n$$ with each of the $$k$$-dimensional coordinate planes is supposed to be a copy of $$\Omega_k$$. A set $$\Omega_n$$ is called admissible in $$L$$ if its intersection with $$L$$ is $$\pm E$$. Given any such set, one may speak of $$d(L)$$ as of the defect of the admissible set $$\Omega_n$$ in $$L$$, and denote this by $$d(\Omega_n,L)$$. There are various classes of popular examples, e.g. the hyperbolic crosses and the Minkowski set $${\mathcal B}_{n,\alpha}: |x_1|^\alpha+\cdots+|x_n|^\alpha\leq 1$$ $$(\alpha>0)$$.
Now let $$d(\Omega_n)=\max_L d(\Omega_n,L)$$ and $$d^*(\Omega_n)=\max_{L'}d(\Omega_n,L')$$, where the first maximum is taken over all lattices of the kind described, and the second maximum is taken over the restricted class of lattices which are obtained from $$\mathbb Z^n$$ by adjoining a single (rational) point. The author presents some asymptotic bounds for these quantities (partly involving octahedra $${\mathcal B}_{n,1}$$ and balls $${\mathcal B}_{n,2}$$) which can be derived from his earlier papers [Mat. Sb., Nov. Ser. 189, No. 6, 117–141 (1998; Zbl 0926.52019); English transl. in Sb. Math. 189, 931–954 (1998); see also N. G. Moshchevitin, Mat. Zametki 58, 558–568 (1995; Zbl 0855.52005) and the author, Grazer Math. Ber. 338, 31–62 (1999; Zbl 0973.11068)].
The main result here is concerned with lower estimates of $$d^*(\Omega_n)$$ for certain subsequences of dimensions. In contrast to the methods used in the papers mentioned, the present proof is of a probabilistic nature.

##### MSC:
 11H16 Nonconvex bodies 11H31 Lattice packing and covering (number-theoretic aspects) 52C17 Packing and covering in $$n$$ dimensions (aspects of discrete geometry)
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