Geodesic
Inhomogeneous Diophantine approximation on curves with non-monotonic error function
Dalʹnevostočnyj matematičeskij žurnal, Tome 13 (2013) no. 2, pp. 164-178.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we prove a convergent part of inhomogeneous Groshev type theorem for non–degenerate curves in Euclidean space where an error function is not necessarily monotonic. Our result naturally incorporates and generalizes the homogeneous measure theorem for non-degenerate curves. In particular, the method of Inhomogeneous Transference Principle and Sprindzuk's method of essential and inessential domains are used in the proof. 
@article{DVMG_2013_13_2_a0,
     author = {N. V. Budarina},
     title = {Inhomogeneous {Diophantine} approximation on curves with non-monotonic error function},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {164--178},
     publisher = {mathdoc},
     volume = {13},
     number = {2},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2013_13_2_a0/}
}
TY  - JOUR
AU  - N. V. Budarina
TI  - Inhomogeneous Diophantine approximation on curves with non-monotonic error function
JO  - Dalʹnevostočnyj matematičeskij žurnal
PY  - 2013
SP  - 164
EP  - 178
VL  - 13
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DVMG_2013_13_2_a0/
LA  - en
ID  - DVMG_2013_13_2_a0
ER  - 
%0 Journal Article
%A N. V. Budarina
%T Inhomogeneous Diophantine approximation on curves with non-monotonic error function
%J Dalʹnevostočnyj matematičeskij žurnal
%D 2013
%P 164-178
%V 13
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DVMG_2013_13_2_a0/
%G en
%F DVMG_2013_13_2_a0
N. V. Budarina. Inhomogeneous Diophantine approximation on curves with non-monotonic error function. Dalʹnevostočnyj matematičeskij žurnal, Tome 13 (2013) no. 2, pp. 164-178. http://geodesic.mathdoc.fr/item/DVMG_2013_13_2_a0/

[1] V. V. Beresnevich, V. I. Bernik, “On a metrical theorem of W. Schmidt”, Acta Arith., 75:3 (1996), 219–233 | MR | Zbl

[2] D. Badziahin, V. V. Beresnevich, S. Velani, “Inhomogeneous theory of dual Diophantine approximation on manifolds”, Adv. Math., 232:1 (2013), 1–35 | DOI | MR | Zbl

[3] V. V. Beresnevich, “A Groshev type theorem for convergence on manifolds”, Acta Math. Hungar., 94 (2002), 99–130 | DOI | MR | Zbl

[4] V. V. Beresnevich, V. I. Bernik, D. Y. Kleinbock, G. A. Margulis, “Metric Diophantine approximation: the Khintchine-Groshev theorem for nondegenerate manifolds”, Mosc. Math. J., 2 (2002), 203–225 | MR | Zbl

[5] V. V. Beresnevich, “On a theorem of V. Bernik in the metric theory of Diophantine approximation”, Acta Arith., 117 (2005), 71–80 | DOI | MR | Zbl

[6] V. V. Beresnevich, S. Velani, “An inhomogeneous transference principle and Diophantine approximation”, Proc. Lond. Math. Soc., 101 (2010), 821–851 | DOI | MR | Zbl

[7] V. I. Bernik, D. Dickinson, M. Dodson, “Approximation of real numbers by values of integer polynomials”, Dokl. Nats. Akad. Nauk Belarusi, 42 (1998), 51–54 | MR | Zbl

[8] V. I. Bernik, D. Y. Kleinbock, G. A. Margulis, “Khintchine–type theorems on manifolds: the convergence case for standard and multiplicative versions”, Internat. Res. Notices, 9 (2001), 453–486 | DOI | MR | Zbl

[9] N. Budarina, D. Dickinson, “Diophantine approximation on non–degenerate curves with non–monotonic error function”, Bull. Lond. Math. Soc., 41 (2009), 137–146 | DOI | MR | Zbl

[10] D. Y. Kleinbock, G. A. Margulis, “Flows on homogeneous spaces and Diophantine approximation on manifolds”, Ann. of Math., 148 (1998), 339–360 | DOI | MR | Zbl

[11] A. Piartly, “Diophantine approximations on submanifolds of Euclidean space”, Funktsional. Anal. i Prilozhen., 3:4 (1969), 59–62 | MR

[12] V. G. Sprindzuk, Mahler's Problem in Metric Number Theory, Transl. Math. Monogr., 25, Amer. Math. Soc., Providence, RI, 1969 | MR