Algebraic lattice in a metric space lattices

E. N. Smirnova ; O. A. Pikhtilkova ; N. N. Dobrovol'skii ; N. M. Dobrovol'skii

Čebyševskij sbornik, Tome 18 (2017) no. 4, pp. 326-338

Résumé

In this article we give a new General definition of an algebraic lattice. It is proved that any rational transformation of algebraic lattices is again an algebraic lattice. It is shown that the reciprocal lattice to algebraic lattices will also be an algebraic lattice, corresponding to a purely-real algebraic field $F_s$ over the rationals $\mathbb{Q}$. Following B. F. Skubenko, we study the fundamental system of pure-real algebraic fields $F_s$ over the rationals $\mathbb{Q}$. Shows the relationship between fundamental systems of algebraic numbers and algebraic lattices. We prove estimates for the norms of transition matrices from an arbitrary nondegenerate matrix for approximating rational matrix. Using the Lemma about the estimation of the norm of the matrix of transition and inverse transition matrices, linking an arbitrary non-degenerate matrix and a nondegenerate approximating the rational matrix, it is shown that the set of algebraic lattices is everywhere dense in a metric space lattices. The theorem is a special case of a more General theorem that for any lattice $\Lambda\in PR_s$ the set of all rational lattices associated with a lattice $\Lambda$ is everywhere dense in $PR_s$. The analogue of this theorem is the assertion that for an arbitrary point of the General clause of $\mathbb{R}^s$, the corresponding $s$-dimensional rational arithmetic space is everywhere dense in $s$-dimensional real arithmetical space $\mathbb{R}^s$.

Détail

Keywords: algebraic lattices, a metric space lattices.