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.
Such a lattice is called critical for K. For its existence, see [10, p. 187].
References
Armitage, J.V.: On a method of Mordell in the geometry of numbers. Mathematika 2, 132–140 (1955)
Blichfeldt, H.: A new principle in the geometry of numbers, with some applications. Trans. Am. Math. Soc. 15, 227–235 (1914)
Bombieri, E.: Sulla dimonstrazione di C.L. Siegel del teorema fondamentale di Minkowski nella geometria dei numeri. Boll. Un. Mat. Ital. (3) 17, 283–288 (1962)
Cassels, J.W.S.: Simultaneous Diophantine approximation. J. Lond. Math. Soc. 30, 119–121 (1955)
Davenport, H.: On the product of three homogeneous linear forms. J. Lond. Math. Soc. 13, 139–145 (1938)
Davenport, H.: On the minimum of a ternary cubic form. J. Lond. Math. Soc. 19, 13–18 (1944)
Davenport, H.: Simultaneous Diophantine approximation. Proc. Lond. Math. Soc. (3) 2, 406–416 (1952)
Davenport, H.: On a theorem of Furtwängler. J. Lond. Math. Soc. 30, 185–195 (1955)
Davenport, H., Mahler, K.: Simultaneous Diophantine approximation. Duke Math. J. 13, 105–111 (1946)
Gruber, P.M., Lekkerkerker, C.G.: Geometry of Numbers. North Holland, Amsterdam (1987)
Hlawka, E., Schoißengeier, J., Taschner, R.: Geometrische und analytische Zahlentheorie. Manz, Wien (1986)
Mordell, L.J.: The product of three homogeneous linear ternary forms. J. Lond. Math. Soc. 17, 107–115 (1942)
Mordell, L.J.: Observation on the minimum of a positive quadratic form in eight variables. J. Lond. Math. Soc. 19, 3–6 (1944)
Mordell, L.J.: On the minimum of a ternary cubic form. J. Lond. Math. Soc. 19, 6–12 (1944)
Mack, J.M.: Simultaneous Diophantine approximation. J. Aust. Math. Soc. Ser. A 24, 266–285 (1977)
Mullender, P.: Lattice points in non-convex regions I. Proc. Kon. Ned. Akad. Wet. 51, 874–884 (1948)
Mullender, P.: Simultaneous approximation. Ann. Math. 52, 417–426 (1950)
Niven, I., Zuckerman, H.S.: Einführung in die Zahlentheorie, Bibliograph. Inst., Mannheim (1975)
Nowak, W.G.: A note on simultaneous Diophantine approximation. Manuscr. Math. 36, 33–46 (1981)
Nowak, W.G.: Proceedings Österr.-Ung.-Slow. Kolloquium über Zahlentheorie, Graz-Mariatrost 1992. Grazer Math. Ber. 318, 105–110 (1992)
Nowak, W.G.: The critical determinant of the double paraboloid and Diophantine approximation in \(\mathbb{R}^{3}\) and \(\mathbb{R}^{4}\). Math. Pannonica 10, 111–122 (1999)
Nowak, W.G.: Diophantine approximation in \(\mathbb{R}^{s}\): on a method of Mordell and Armitage. In: Algebraic Number Theory and Diophantine Analysis. Proceedings of the Conference Held in Graz, pp. 339–349, 30 August–5 September 1998. W. de Gruyter, Berlin (2000)
Nowak, W.G.: Lower bounds for simultaneous Diophantine approximation constants. Commun. Math. 22 (1), 71–76 (2014)
Pohst, M., Zassenhaus, H.: Algorithmic Algebraic Number Theory. Encyclopedia of Mathematics and its Applications, vol. 30. Cambridge University Press, Cambridge (1989)
Prasad, A.V.: Simultaneous Diophantine approximation. Proc. Indian Acad. Sci. A 31, 1–15 (1950)
Siegel, C.L.: Über Gitterpunkte in konvexen Körpern und ein damit zusammenhängendes Extremalproblem. Acta Math. 65, 307–335 (1935)
Spohn, W.G.: Blichfeldt’s theorem and simultaneous Diophantine approximation. Am. J. Math. 90, 885–894 (1968)
Žilinskas, G.: On the product of four homogeneous linear forms. J. Lond. Math. Soc. 16, 27–37 (1941)
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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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