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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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References

1.Adiceam, F., An extension of a theorem of Duffin and Schaeffer in Diophantine approximation, Acta Arith. 162(3) (2014), 243–254.Google Scholar
2.Budarina, N., Dickinson, D. and Levesley, J., Simultaneous Diophantine approximation on polynomial curves, Mathematika 56(1) (2010), 77–85.CrossRefGoogle Scholar
3.Dickinson, D., Ideas and results from the theory of Diophantine approximation, in Diophantine phenomena in differential equations and dynamical systems (Research Institute for Mathematical Sciences, Kyoto, 2004).Google Scholar
4.Dickinson, H. and Dodson, M. M., Simultaneous Diophantine approximation on the circle and Hausdorff dimension, Math. Proc. Camb. Phil. Soc. 130(3) (2001), 515–522.Google Scholar
5.Duffin, R. J. and Schaeffer, A. C., Khintchine's problem in metric Diophantine approximation, Duke Math. J. 8 (1941), 243–255.CrossRefGoogle Scholar
6.Falconer, K. J., The geometry of fractal sets, Cambridge Tracts in Mathematics, Volume 85, p. 14 (Cambridge University Press, 1985).Google Scholar
7.Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, 4th edn (Clarendon, Oxford, 1960).Google Scholar
8.Korolev, M. A., On the average number of power residues modulo a composite number, Izv. Math. 74(6) (2010), 1225–1254.Google Scholar
9.Rynne, B. P., Hausdorff dimension and generalized simultaneous Diophantine approximation, Bull. Lond. Math. Soc. 30(4) (1998), 365–376.Google Scholar
10.Sprindžuk, V. G., Metric theory of Diophantine approximations (transl. Silverman, R. A.), Scripta Series in Mathematics (V. H. Winston, Washington D.C., 1979).Google Scholar
11.Vaughan, R. C. and Velani, S., Diophantine approximation on planar curves: the convergence theory, Invent. Math. 160(1) (2006), 103–124.Google Scholar
12.Wiles, A., Modular elliptic curves and Fermat's last theorem, Annals Math. (2) 142 (1995), 443–551.Google Scholar