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Vertical Shift and Simultaneous Diophantine Approximation on Polynomial Curves

Published online by Cambridge University Press:  27 October 2014

Faustin Adiceam
Faustin Adiceam*
Affiliation:
Department of Mathematics, Logic House, National University of Ireland, Maynooth, Ireland, [email protected])
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Abstract

The Hausdorff dimension of the set of simultaneously τ-well-approximable points lying on a curve defined by a polynomial P(X) + α, where P(X) ∈ ℤ[X] and α ∈ ℝ, is studied when τ is larger than the degree of P(X). This provides the first results related to the computation of the Hausdorff dimension of the set of well-approximable points lying on a curve that is not defined by a polynomial with integer coefficients. The proofs of the results also include the study of problems in Diophantine approximation in the case where the numerators and the denominators of the rational approximations are related by some congruential constraint.

Keywords

diophantine approximationmanifoldspolynomial curvesmetric theoryPrimary 11J8311K60
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 1 , February 2015 , pp. 1 - 26
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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