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Diophantine approximation on polynomial curves

Published online by Cambridge University Press:  09 February 2017

JOHANNES SCHLEISCHITZ
JOHANNES SCHLEISCHITZ*
Affiliation:
University of Ottawa, Dept. of Mathematics and Statistics, 585 King Edward Ave Ottawa, ON K1N 6N5, Canada. e-mail: [email protected]
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Abstract

In a paper from 2010, Budarina, Dickinson and Levesley studied the rational approximation properties of curves parametrised by polynomials with integral coefficients in Euclidean space of arbitrary dimension. Assuming the dimension is at least three and excluding the case of linear dependence of the polynomials together with P(X) ≡ 1 over the rational number field, we establish proper generalisations of their main result.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 3 , November 2017 , pp. 533 - 546
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Baker, R. C. Dirichlet's theorem on Diophantine approximation. Math. Proc. Camb. Phil. Soc. 83 (1978), 37–59.CrossRefGoogle Scholar
[2] Beresnevich, V., Dickinson, D. and Velani, S. L.. Diophantine approximation on planar curves and the distribution of rational points. with an appendix by Vaughan, R.C.: ‘Sums of two squares near perfect squares’. Ann. of Math. 166 (2007), 367–426.Google Scholar
[3] Bernik, V. I. An application of Hausdorff dimension in the theory of Diophantine approximation. Acta Arith. 42 (1983), 219–253 (in Russian).Google Scholar
[4] Budarina, N., Dickinson, D. and Levesley, J.. Simultaneous Diophantine approximation on polynomial curves. Mathematika 56 (2010), 77–85.Google Scholar
[5] Bugeaud, Y. On simultaneous rational approximation to a real numbers and its integral powers. Ann. Inst. Fourier (Grenoble) (6) 60 (2010), 2165–2182.Google Scholar
[6] Bugeaud, Y. and Laurent, M.. Exponents in Diophantine approximation. Diophantine Geometry Proceedings, Scuola Normale Superiore Pisa, Ser. CRM, 4 (2007), 101–121.Google Scholar
[7] Dodson, M. M., Rynne, B. P. and Vickers, J. A. G.. Metric Diophantine approximation and Hausdorff dimension on manifolds. Math. Proc. Camb. Phil. Soc. 130 (2001), 515–522.Google Scholar
[8] Jarník, V. Über die simultanen Diophantische Approximationen. Math. Z. 33 (1931), 505–543.CrossRefGoogle Scholar
[9] Schleischitz, J. On the spectrum of Diophantine approximation constants. Mathematika 62 (2016), no. 1, 79–100.CrossRefGoogle Scholar
[10] Schleischitz, J. Generalizations of a result of Jarnik on simultaneous approximation. To appear in Mosc. J. Comb. Number Theory arXiv: 1410.6697.Google Scholar
[11] Vaughan, R. C. and Velani, S. L.. Diophantine approximation on planar curves: the convergence theory. Invent. Math. 166 (2006), 103–124.Google Scholar