Geodesic
Simplex–karyon algorithm of multidimensional continued fraction expansion
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic number theory, Tome 299 (2017), pp. 283-303

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

A simplex–karyon algorithm for expanding real numbers $\alpha =(\alpha _1,\dots ,\alpha _d)$ in multidimensional continued fractions is considered. The algorithm is based on a $(d+1)$-dimensional superspace $\mathbf S$ with embedded hyperplanes: a karyon hyperplane $\mathbf K$ and a Farey hyperplane $\mathbf F$. The approximation of numbers $\alpha $ by continued fractions is performed on the hyperplane $\mathbf F$, and the degree of approximation is controlled on the hyperplane $\mathbf K$. A local $\wp (r)$-strategy for constructing convergents is chosen, with a free objective function $\wp (r)$ on the hyperplane $\mathbf K$.
Keywords: multidimensional continued fractions, best approximations, Farey sums. 
V. G. Zhuravlev. Simplex–karyon algorithm of multidimensional continued fraction expansion. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analytic number theory, Tome 299 (2017), pp. 283-303. http://geodesic.mathdoc.fr/item/TM_2017_299_a16/
@article{TM_2017_299_a16,
     author = {V. G. Zhuravlev},
     title = {Simplex{\textendash}karyon algorithm of multidimensional continued fraction expansion},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {283--303},
     year = {2017},
     volume = {299},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2017_299_a16/}
}
TY  - JOUR
AU  - V. G. Zhuravlev
TI  - Simplex–karyon algorithm of multidimensional continued fraction expansion
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2017
SP  - 283
EP  - 303
VL  - 299
UR  - http://geodesic.mathdoc.fr/item/TM_2017_299_a16/
LA  - ru
ID  - TM_2017_299_a16
ER  - 
%0 Journal Article
%A V. G. Zhuravlev
%T Simplex–karyon algorithm of multidimensional continued fraction expansion
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2017
%P 283-303
%V 299
%U http://geodesic.mathdoc.fr/item/TM_2017_299_a16/
%G ru
%F TM_2017_299_a16

[1] Arnoux P., Labbé S., On some symmetric multidimensional continued fraction algorithms, E-print, 2015, arXiv: 1508.07814 [math.DS]

[2] Berthé V., Labbé S., “Factor complexity of $S$-adic words generated by the Arnoux–Rauzy–Poincaré algorithm”, Adv. Appl. Math., 63 (2015), 90–130 | DOI | MR | Zbl

[3] Treizième Congrès des mathématiciens scandinaves (Helsinki, 1957), Mercators Tryckeri, Helsinki, 1958, 45–64

[4] Cassaigne J., Un algorithme de fractions continues de complexité linéaire, Talk at the Dyna3S Meeting (Paris, Oct. 12, 2015)

[5] Grabiner D.J., “Farey nets and multidimensional continued fractions”, Monatsh. Math., 114:1 (1992), 35–61 | DOI | MR | Zbl

[6] Mönkemeyer R., “Über Fareynetze in $n$ Dimensionen”, Math. Nachr., 11 (1954), 321–344 | DOI | MR | Zbl

[7] Nogueira A., “The three-dimensional Poincaré continued fraction algorithm”, Isr. J. Math., 90:1–3 (1995), 373–401 | DOI | MR | Zbl

[8] W. M. Schmidt, Diophantine Approximation, Lect. Notes Math., 785, Springer, Berlin, 1980 | MR | MR | Zbl

[9] Schweiger F., Multidimensional continued fractions, Oxford Univ. Press, Oxford, 2000 | MR | Zbl

[10] Selmer E.S., “Continued fractions in several dimensions”, Nordisk Mat. Tidskr., 9 (1961), 37–43 | MR | Zbl

[11] V. G. Zhuravlev, “Two-dimensional approximations by the method of dividing toric tilings”, J. Math. Sci., 217:1 (2016), 54–64 | DOI | MR | Zbl

[12] V. G. Zhuravlev, “Differentiation of induced toric tiling and multidimensional approximations of algebraic numbers”, J. Math. Sci., 222:5 (2017), 544–584 | DOI | MR | Zbl

[13] V. G. Zhuravlev, “Simplex-module algorithm for expansion of algebraic numbers in multidimensional continued fractions”, J. Math. Sci., 225:6 (2017), 924–949 | DOI | MR | Zbl

[14] V. G. Zhuravlev, “Periodic karyon expansions of cubic irrationals in continued fractions”, Proc. Steklov Inst. Math., 296, Suppl. 2 (2017), 36–60 | DOI | DOI | MR