Abstract
In the present paper, we prove an analog of Khinchin's metric theorem in the case of linear Diophantine approximations of plane curves defined over the ring of \(p\)-adic integers by means of (Mahler) normal functions. We also prove some general assertions needed to generalize this result to the case of spaces of higher dimension.
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REFERENCES
A. Khintchine, “Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen,” Math. Ann., 92 (1924), 115–125.
K. Mahler, “Ñber das Maβ der Menge aller S-Zahlen,” Math. Ann., 106 (1932), 131–139.
V. G. Sprindzhuk, Mahler's Problem in Metric Number Theory [in Russian], Nauka i Tekhnika, Minsk, 1967; English translation in: Amer. Math. Soc., Providence, R.I., 1969.
V. I. Bernik, “The exact order of approximating zero by values of integral polynomials” [in Russian], Acta Arith., 53 (1989), no. 1, 17–28.
V. Beresnevich, “On approximation of real numbers by real algebraic numbers,” Acta Arith., 50 (1999), no. 2, 97–112.
V. I. Bernik and M. M. DodsonMetric Diophantine Approximation on Manifolds, Cambridge Univ. Press, Cambridge, 1999.
V. G. Sprindzhuk, Metric Theory of Diophantine Approximations [in Russian], Nauka, Moscow, 1977; English translation in: Winston, Washington, D.C., 1979.
V. G. Sprindzhuk, “Achievements and problems of the theory of Diophantine approximations,” Uspekhi Mat. Nauk [Russian Math. Surveys], 35 (1980), no. 4, 3–68.
D. Y. Kleinbock and G. A. Margulis, “Flows on homogeneous spaces and Diophantine approximation on manifolds,” Ann. Math., 148 (1998), 339–360.
K. Mahler, “Ñber transcendente p-adische Zahlen,” Compos. Math., 2 (1935), 259–275.
W. W. Adams, “Transcendental numbers in the p-adic domain,” Amer. J. Math., 88 (1966), no. 2, 279–308.
K. Mahler, p-Adic Numbers and Their Functions, vol. 76, Cambridge Tracts in Mathematics, Cambridge Univ. Press, Cambridge–New York, 1981.
Yu. V. Melnichuk, “On the metric theory of simultaneous Diophantine approximations of p-adic numbers,” Dokl. Akad. Nauk Ukrain. SSR Ser. A (1978), no. 5, 394–397.
V. Beresnevich, V. Bernik, M. Dodson, and H. Dickinson, “The Khintchine–Groshev theorem for lanar curves,” Proc. Royal Soc. London. Ser. A, 455 (1999), 3053–3063.
V. Bernik, M. Dodson, and H. Dickinson, “A Khintchine type version of Schmidt's theorem for planar curves,” Proc. Royal Soc. London. Ser. A, 454 (1998), 179–185.
É. I. Kovalevskaya, “A metric theorem on the exact order of approximation of zero by values of integer polynomials in Q,” Dokl. Nats. Akad. Nauk Belarusi, 43 (1999), no. 5, 34–36.
É. I. Kovalevskaya, “A p-adic version of Khinchin's theorem for plane curves in the case of convergence,” Dokl. Nats. Akad. Nauk Belarusi, 44 (2000), no. 2, 28–30.
V. V. Beresnevich, “Application of the concept of regular systems of points in metric number theory,” Vestsi Nats. Akad. Navuk Belarusi. Ser. Fiz.-Mat. Navuk (2000), no. 1, 35–39.
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Beresnevich, V.V., Kovalevskaya, É.I. On Diophantine Approximations of Dependent Quantities in the p-adic Case. Mathematical Notes 73, 21–35 (2003). https://doi.org/10.1023/A:1022165815830
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DOI: https://doi.org/10.1023/A:1022165815830