Geodesic
A NOTE ON SIMULTANEOUS AND MULTIPLICATIVE DIOPHANTINE APPROXIMATION ON PLANAR CURVES
Glasgow mathematical journal, Tome 49 (2007) no. 2, pp. 367-375

Voir la notice de l'article provenant de la source Cambridge University Press

Let be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in with two independent approximation functions; that is if a certain sum converges then the set of all points (x,y) on the curve which satisfy simultaneously the inequalities ||qx|| < ψ1(q) and ||qy|| < ψ2(q) infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for multiplicative approximation ||qx|| ||qy|| < ψ(q) we establish a Hausdorff measure convergence result for the same class of curves, the first such result for a general class of manifolds in this particular setup.
DOI : 10.1017/S0017089507003722
Mots-clés : Primary 11J83, Secondary 11J13, 11K60 
BADZIAHIN, DZMITRY; LEVESLEY, JASON. A NOTE ON SIMULTANEOUS AND MULTIPLICATIVE DIOPHANTINE APPROXIMATION ON PLANAR CURVES. Glasgow mathematical journal, Tome 49 (2007) no. 2, pp. 367-375. doi: 10.1017/S0017089507003722
@article{10_1017_S0017089507003722,
     author = {BADZIAHIN, DZMITRY and LEVESLEY, JASON},
     title = {A {NOTE} {ON} {SIMULTANEOUS} {AND} {MULTIPLICATIVE} {DIOPHANTINE} {APPROXIMATION} {ON} {PLANAR} {CURVES}},
     journal = {Glasgow mathematical journal},
     pages = {367--375},
     year = {2007},
     volume = {49},
     number = {2},
     doi = {10.1017/S0017089507003722},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089507003722/}
}
TY  - JOUR
AU  - BADZIAHIN, DZMITRY
AU  - LEVESLEY, JASON
TI  - A NOTE ON SIMULTANEOUS AND MULTIPLICATIVE DIOPHANTINE APPROXIMATION ON PLANAR CURVES
JO  - Glasgow mathematical journal
PY  - 2007
SP  - 367
EP  - 375
VL  - 49
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089507003722/
DO  - 10.1017/S0017089507003722
ID  - 10_1017_S0017089507003722
ER  - 
%0 Journal Article
%A BADZIAHIN, DZMITRY
%A LEVESLEY, JASON
%T A NOTE ON SIMULTANEOUS AND MULTIPLICATIVE DIOPHANTINE APPROXIMATION ON PLANAR CURVES
%J Glasgow mathematical journal
%D 2007
%P 367-375
%V 49
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089507003722/
%R 10.1017/S0017089507003722
%F 10_1017_S0017089507003722

[1] 1. Beresnevich, V., Dickinson, D. and Velani, S., Measure theoretic laws for lim sup sets, Mem. Amer. Math. Soc. 179 (2006), no. 846. Google Scholar

[2] 2. Beresnevich, V., Dickinson, H. and Velani, S.. Diophantine approximation on planar curves and the distribution of rational points, with an Appendix, Sums of two squares near perfect squares, by Vaughan, R. C., Ann Math., to appear: Preprint (53pp) arXiv:math.NT/0401148. Google Scholar

[3] 3. Beresnevich, V. and Velani, S., A note on simultaneous Diophantine approximation on planar curves. Preprint (23pp) arXiv:math.NT/0412141. Google Scholar

[4] 4. Falconer, K., Fractal geometry: mathematical foundations and applications (Wiley, 1990). Google Scholar

[5] 5. Falconer, K., Techniques in fractal geometry (Wiley, 1997). Google Scholar

[6] 6. Mattila, P., Geometry of sets and measures in Euclidean space, Cambridge Studies in Advanced Mathematics, No. 44, (Cambridge University Press, 1995). Google Scholar | DOI

[7] 7. Vaughan, R. C. and Velani, S., Diophantine approximation on planar curves: the convergence theory, Invent. Math 166 (2006), 103–124. Google Scholar

Cité par Sources :