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Simultaneous Diophantine Approximation: Searching for Analogues of Hurwitz’s Theorem

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Abstract

Hurwitz’s classic theorem of Diophantine approximation tells us that for any irrational number α, there exist infinitely many reduced fractions p∕q so that \(\vert \alpha - p/q\vert < (\sqrt{5}\,q^{2})^{-1}\) and that this is no longer true if \(\sqrt{5}\) is replaced by some larger constant. Attempting to generalize this to dimensions s ≥ 2, one is concerned with the problem to determine, resp., to estimate the supremum θ of all reals c so that, for every real but not all rational s-tuple α, there exist infinitely many \(\mathbf{p} \in \mathbb{Z}^{s}\) and positive integers q satisfying gcd(p, q) = 1 and\(\vert \alpha - q^{-1}\mathbf{p}\vert < (q\sqrt[s]{cq})^{-1}\), where | ⋅ | is any norm in \(\mathbb{R}^{s}\). This survey focuses on the cases of the maximum and the Euclidean norms, giving a survey on the most relevant methods and results on these constants θ.

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Notes

  1. 1.

    Such a lattice is called critical for K. For its existence, see [10, p. 187].

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Authors and Affiliations

  1. Institute of Mathematics, DIBB, BOKU Wien, 1180, Vienna, Austria

    Werner Georg Nowak

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  1. Werner Georg Nowak
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Correspondence to Werner Georg Nowak .

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Editors and Affiliations

  1. Department of Mathematics, National Technical Unversity of Athens, Athens, Greece

    Themistocles M. Rassias

  2. University of Florida, GAINESVILLE, Florida, USA

    Panos M. Pardalos

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Nowak, W.G. (2016). Simultaneous Diophantine Approximation: Searching for Analogues of Hurwitz’s Theorem. In: Rassias, T., Pardalos, P. (eds) Essays in Mathematics and its Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-31338-2_9

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