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Zap. Nauchn. Sem. POMI, 2016, Volume 445, Pages 33–92 (Mi znsl6275)  

This article is cited in 11 scientific papers (total in 11 papers)

Differentiation of induced toric tilings and multi-dimensional approximations of algebraic numbers

V. G. Zhuravlev

Vladimir State University, Vladimir, Russia

Abstract: We consider the induced tilings $\mathcal{T=T}|_\mathrm{Kr}$ of the $D$-dimensional torus $\mathbb T^D$ generated by embedded karyons $\mathrm{Kr}$. The differentiations $\sigma\colon\mathcal{T\to T}^\sigma$ are defined under which we obtaine again the induced tilings $\mathcal T^\sigma=\mathcal T|_{\mathrm{Kr}^\sigma}$ with a derivative karyon $\mathrm{Kr}^\sigma$. They are used for approximation of $0\in\mathbb T^D$ by an infinite sequence of points $x_j\equiv j\alpha\mod\mathbb Z^D$ for $j=0,1,2,…$, where $\alpha=(\alpha_1,…,\alpha_D)$ is vector whose coordinates $\alpha_1,…,\alpha_D$ belong to an algebraic field $\mathbb Q(\theta)$ of degree $D+1$ over the rational field $\mathbb Q$. For this purpose, we construct an infinite sequence of convex parallelohedra $T^{(i)}\subset\mathbb T^D$ for $i=0,1,2,…$ and define for them some natural oders $m^{(0)}<m^{(1)}<…<m^{(i)}<…$ Then the above parallelohedra contain a subsequence of points $\{x_{j'}\}_{j'=1}^\infty$ that give the best approximation of $0\in\mathbb T^D$.

Key words and phrases: toric exchange, induced decomposition, best multi-dimensional approximations.

Funding Agency Grant Number
Russian Science Foundation 14-11-00433


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English version:
Journal of Mathematical Sciences (New York), 2017, 222:5, 544–584

Bibliographic databases:

UDC: 511
Received: 16.01.2016

Citation: V. G. Zhuravlev, “Differentiation of induced toric tilings and multi-dimensional approximations of algebraic numbers”, Analytical theory of numbers and theory of functions. Part 31, Zap. Nauchn. Sem. POMI, 445, POMI, St. Petersburg, 2016, 33–92; J. Math. Sci. (N. Y.), 222:5 (2017), 544–584

Citation in format AMSBIB
\Bibitem{Zhu16}
\by V.~G.~Zhuravlev
\paper Differentiation of induced toric tilings and multi-dimensional approximations of algebraic numbers
\inbook Analytical theory of numbers and theory of functions. Part~31
\serial Zap. Nauchn. Sem. POMI
\yr 2016
\vol 445
\pages 33--92
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6275}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3511159}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2017
\vol 222
\issue 5
\pages 544--584
\crossref{https://doi.org/10.1007/s10958-017-3321-8}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85015629041}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. G. Zhuravlev, “Bounded remainder sets”, J. Math. Sci. (N. Y.), 222:5 (2017), 585–640  mathnet  crossref  mathscinet
    2. V. G. Zhuravlev, “Periodic karyon expansions of cubic irrationals in continued fractions”, Proc. Steklov Inst. Math., 296, suppl. 2 (2017), 36–60  mathnet  crossref  crossref  isi  elib
    3. V. G. Zhuravlev, “Periodic karyon expansions of algebraic units in multidimensional continued fractions”, J. Math. Sci. (N. Y.), 225:6 (2017), 893–923  mathnet  crossref  mathscinet
    4. V. G. Zhuravlev, “Simplex-module algorithm for expansion of algebraic numbers in multidimensional continued fractions”, J. Math. Sci. (N. Y.), 225:6 (2017), 924–949  mathnet  crossref  mathscinet
    5. V. G. Zhuravlev, “Karyon expansions of Pisot numbers in multidimensional continued fractions”, J. Math. Sci. (N. Y.), 225:6 (2017), 950–968  mathnet  crossref  mathscinet
    6. V. G. Zhuravlev, “Simplex–karyon algorithm of multidimensional continued fraction expansion”, Proc. Steklov Inst. Math., 299 (2017), 268–287  mathnet  crossref  crossref  isi  elib
    7. V. G. Zhuravlev, “Drobno-lineinaya invariantnost mnogomernykh tsepnykh drobei”, Analiticheskaya teoriya chisel i teoriya funktsii. 33, Posvyaschaetsya pamyati Galiny Vasilevny KUZMINOI, Zap. nauchn. sem. POMI, 458, POMI, SPb., 2017, 42–76  mathnet
    8. V. G. Zhuravlev, “Drobno-lineinaya invariantnost simpleks-modulnogo algoritma razlozheniya algebraicheskikh chisel v mnogomernye tsepnye drobi”, Analiticheskaya teoriya chisel i teoriya funktsii. 33, Posvyaschaetsya pamyati Galiny Vasilevny KUZMINOI, Zap. nauchn. sem. POMI, 458, POMI, SPb., 2017, 77–103  mathnet
    9. V. G. Zhuravlev, “Yadernyi algoritm razlozheniya v mnogomernye tsepnye drobi”, Algebra i teoriya chisel. 1, Posvyaschaetsya pamyati Olega Mstislavovicha FOMENKO, Zap. nauchn. sem. POMI, 469, POMI, SPb., 2018, 32–63  mathnet
    10. V. G. Zhuravlev, “Unimodulyarnost indutsirovannykh razbienii tora”, Algebra i teoriya chisel. 1, Posvyaschaetsya pamyati Olega Mstislavovicha FOMENKO, Zap. nauchn. sem. POMI, 469, POMI, SPb., 2018, 64–95  mathnet
    11. V. G. Zhuravlev, “Unimodulyarnaya invariantnost yadernykh razlozhenii algebraicheskikh chisel v mnogomernye tsepnye drobi”, Algebra i teoriya chisel. 1, Posvyaschaetsya pamyati Olega Mstislavovicha FOMENKO, Zap. nauchn. sem. POMI, 469, POMI, SPb., 2018, 96–137  mathnet
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