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 Chebyshevskii Sb., 2017, Volume 18, Issue 4, Pages 326–338 (Mi cheb615)

Algebraic lattice in a metric space lattices

E. N. Smirnovaa, O. A. Pikhtilkovaa, N. N. Dobrovol'skiib, N. M. Dobrovol'skiic

a Orenburg State University
b Tula State University
c Tula State Pedagogical University

Abstract: 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 $\mathbb{R}^s$.

Keywords: algebraic lattices, a metric space lattices.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-41-710194_ð_öåíòð_à15-01-01540_a

Full text: PDF file (638 kB)
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UDC: 511.42
Accepted:15.12.2017

Citation: E. N. Smirnova, O. A. Pikhtilkova, N. N. Dobrovol'skii, N. M. Dobrovol'skii, “Algebraic lattice in a metric space lattices”, Chebyshevskii Sb., 18:4 (2017), 326–338

Citation in format AMSBIB
\Bibitem{SmiPikDob17} \by E.~N.~Smirnova, O.~A.~Pikhtilkova, N.~N.~Dobrovol'skii, N.~M.~Dobrovol'skii \paper Algebraic lattice in a metric space lattices \jour Chebyshevskii Sb. \yr 2017 \vol 18 \issue 4 \pages 326--338 \mathnet{http://mi.mathnet.ru/cheb615} 

• http://mi.mathnet.ru/eng/cheb615
• http://mi.mathnet.ru/eng/cheb/v18/i4/p326

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This publication is cited in the following articles:
1. S. S. Demidov, E. A. Morozova, V. N. Chubarikov, I. Yu. Rebrova, I. N. Balaba, N. N. Dobrovolskii, N. M. Dobrovolskii, L. P. Dobrovolskaya, A. V. Rodionov, O. A. Pikhtilkova, “Teoretiko-chislovoi metod v priblizhennom analize”, Chebyshevskii sb., 18:4 (2017), 6–85
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