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

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DOI

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
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     title = {A {NOTE} {ON} {SIMULTANEOUS} {AND} {MULTIPLICATIVE} {DIOPHANTINE} {APPROXIMATION} {ON} {PLANAR} {CURVES}},
     journal = {Glasgow mathematical journal},
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     year = {2007},
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