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.

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.

Reviewer: Gerhard Ramharter (Wien)

##### MSC:

11H16 | Nonconvex bodies |

11H31 | Lattice packing and covering (number-theoretic aspects) |

52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |