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A note on simultaneous and multiplicative Diophantine approximation on planar curves

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JournalGlasgow Mathematical Journal
DatePublished - 9 Aug 2007
Issue number2
Volume49
Number of pages9
Pages (from-to)367-375
Original languageEnglish

Abstract

Let C be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in R-2 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 vertical bar vertical bar qx vertical bar vertical bar < psi(1)(q) and vertical bar vertical bar qy vertical bar vertical bar < psi(2)(q) infinitely often has induced measure 0. This completes the metric theory for the Lebesgue case. Further, for multiplicative approximation vertical bar vertical bar qx vertical bar vertical bar vertical bar vertical bar qy vertical bar vertical bar < psi (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.

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