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Recent advances in Diophantine approximation

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Number Theory, Analysis and Geometry

Abstract

A basic question of Diophantine approximation, which is the first issue we discuss, is to investigate the rational approximations to a single real number. Next, we consider the algebraic or polynomial approximations to a single complex number, as well as the simultaneous approximation of powers of a real number by rational numbers with the same denominator. Finally we study generalisations of these questions to higher dimensions. Several recent advances have been made by B. Adamczewski, Y. Bugeaud, S. Fischler, M. Laurent, T. Rivoal, D. Roy, and W.M. Schmidt, among others. We review some of these works.

Mathematics Subject Classification (2010): 11Jxx

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Notes

  1. 1.

    Compare with the definition of singular systems in §7, Chap. V of [46].

  2. 2.

    The occurrence of − 1 in the exponent of the right-hand side of (31) is already plain for degree 1 polynomials, comparing \(\vert \alpha- p/q\vert \) and |qα − p|.

  3. 3.

    One-dimensional Lebesgue measure on Γ.

  4. 4.

    see Beresnevich: Rational points near manifolds and metric Diophantine approximation. Ann. of Math. (to appear). http://arxiv.org/abs/0904.0474.

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Acknowledgements

Many thanks to Boris Adamczewski, Victor Beresnevich, Yann Bugeaud, Maurice Dodson, Michel Laurent, Claude Levesque, Damien Roy for their enlightening remarks and their comments on preliminary versions of this paper. Sections 2.7 and 3.6, as well as part of Section 1.2, have been written by Victor Beresnevich and Maurice Dodson. I wish also to thank Dinakar Ramakrishnan who completed the editorial work in a very efficient way.

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  1. Université Pierre et Marie Curie–Paris 6, UMR 7586 IMJ Institut de, Mathématiques de Jussieu, 4 Place Jussieu, 75252, Paris Cedex 05, France

    Michel Waldschmidt

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Correspondence to Michel Waldschmidt .

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  1. , Department of Mathematics, Columbia University, Broadway 2990, New York, 10027, New York, USA

    Dorian Goldfeld

  2. , Department of Mathematics, City College of New York, New York, 10031, New York, USA

    Jay Jorgenson

  3. , Department of Mathematics, Yale University, Hillhouse Ave. 10, New Haven, 06520, Connecticut, USA

    Peter Jones

  4. , Department of Mathematics, California Institute of Technology, E. California Blvd. 1200, Pasadena, 91125, California, USA

    Dinakar Ramakrishnan

  5. , Department of Mathematics, University of California at Berkeley, Evans Hall 925, Berkeley, 94720-3840, California, USA

    Kenneth Ribet

  6. , Department of Mathematics, Harvard University, Science Center/One Oxford Street, Cambridge, 02138, Massachusetts, USA

    John Tate

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Waldschmidt, M. (2012). Recent advances in Diophantine approximation. In: Goldfeld, D., Jorgenson, J., Jones, P., Ramakrishnan, D., Ribet, K., Tate, J. (eds) Number Theory, Analysis and Geometry. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1260-1_29

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