A note on simultaneous and multiplicative Diophantine approximation on planar curves

D. BADZIAHIN; J. LEVESLEY

doi:10.1017/S0017089507003722

A note on simultaneous and multiplicative Diophantine approximation on planar curves

D. BADZIAHIN, J. LEVESLEY

Mathematics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	367-375
Number of pages	9
Journal	Glasgow Mathematical Journal
Volume	49
Issue number	2
DOIs	https://doi.org/10.1017/S0017089507003722
Publication status	Published - 9 Aug 2007

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10.1017/S0017089507003722

http://dx.doi.org/10.1017/S0017089507003722

Cite this

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title = "A note on simultaneous and multiplicative Diophantine approximation on planar curves",

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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T1 - A note on simultaneous and multiplicative Diophantine approximation on planar curves

AU - BADZIAHIN, D.

AU - LEVESLEY, J.

PY - 2007/8/9

Y1 - 2007/8/9

N2 - 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.

AB - 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.

U2 - 10.1017/S0017089507003722

DO - 10.1017/S0017089507003722

M3 - Article

SN - 1469-509X

VL - 49

SP - 367

EP - 375

JO - Glasgow Mathematical Journal

JF - Glasgow Mathematical Journal

IS - 2

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