Geodesic
Vectors of a given Diophantine type. II
Sbornik. Mathematics, Tome 204 (2013) no. 4, pp. 463-484 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We prove the existence of a family of vectors with continuum many elements $\mathbf v\in\mathbb{R}^s$ admitting infinitely many simultaneous $(\varphi(p)/p^{1/s})(1+B\cdot\varphi^{1+1/s}(p))$-approximations and admitting no simultaneous $(\varphi(p)/p^{1/s})(1-B\cdot\varphi^{1+1/s}(p))$-approximation. We prove that for $0 the closed interval $[t,t(1+16B\cdot t^{1+1/s})]$ contains an element of the $s$-dimensional Lagrange spectrum. Here $A$, $B$ and $T$ stand for some positive constants depending on the dimension $s$ only and $\varphi$ is a positive nonincreasing function of positive integer argument such that $\varphi(1)\le A$. Bibliography: 5 titles.
Keywords: simultaneous Diophantine approximations, Lagrange spectrum, Euclidean space, simultaneous $\psi$-approximation. 
@article{SM_2013_204_4_a0,
     author = {R. K. Akhunzhanov},
     title = {Vectors of a~given {Diophantine} {type.~II}},
     journal = {Sbornik. Mathematics},
     pages = {463--484},
     year = {2013},
     volume = {204},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2013_204_4_a0/}
}
TY  - JOUR
AU  - R. K. Akhunzhanov
TI  - Vectors of a given Diophantine type. II
JO  - Sbornik. Mathematics
PY  - 2013
SP  - 463
EP  - 484
VL  - 204
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_2013_204_4_a0/
LA  - en
ID  - SM_2013_204_4_a0
ER  - 
%0 Journal Article
%A R. K. Akhunzhanov
%T Vectors of a given Diophantine type. II
%J Sbornik. Mathematics
%D 2013
%P 463-484
%V 204
%N 4
%U http://geodesic.mathdoc.fr/item/SM_2013_204_4_a0/
%G en
%F SM_2013_204_4_a0
R. K. Akhunzhanov. Vectors of a given Diophantine type. II. Sbornik. Mathematics, Tome 204 (2013) no. 4, pp. 463-484. http://geodesic.mathdoc.fr/item/SM_2013_204_4_a0/

[1] M. Hall, jr., “On the sum and product of continued fractions”, Ann. of Math. (2), 48:4 (1948), 966–993 | DOI | MR | Zbl

[2] T. W. Cusick, M. E. Flahive, The Markoff and Lagrange spectra, Math. Surveys Monogr., 30, Amer. Math. Soc., Providence, RI, 1989 | MR | Zbl

[3] A. V. Malyshev, “Markov and Lagrange spectra (survey of the literature)”, J. Soviet Math., 16:1 (1981), 767–788 | DOI | MR | Zbl | Zbl

[4] V. Jarnik, “Über die simultanen diophantischen Approximationen”, Math. Z., 33:1 (1931), 505–543 | DOI | MR | Zbl

[5] R. K. Akhundzhanov, N. G. Moshchevitin, “Vectors of given Diophantine type”, Math. Notes, 80:3–4 (2006), 318–328 | DOI | DOI | MR | Zbl