Far Eastern Mathematical Journal

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On regular systems of algebraic p-adic numbers of arbitrary degree in small cylinders

N. Budarina, F. Götze

2015, issue 2, . 133155

In this paper we prove that for any sufficiently large $Q\in{\mathbb N}$ there exist cylinders $K\subset{\mathbb Q}_p$ with Haar measure $\mu(K)\le \frac{1}{2}Q^{-1}$ which do not contain algebraic $p$-adic numbers $\alpha$ of degree $\deg\alpha=n$ and height $H(\alpha)\le Q$. The main result establishes in any cylinder $K$, $\mu(K)>c_1Q^{-1}$, $c_1>c_0(n)$, the existence of at least $c_{3}Q^{n+1}\mu(K)$ algebraic $p$-adic numbers $\alpha\in K$ of degree $n$ and $H(\alpha)\le Q$.

integer polynomials, algebraic p-adic numbers, regular system, Haar measure

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[1] A. Baker, W.M. Schmidt, "Diophantine approximation and Hausdorff dimension", Proc. London Math. Soc., 21 (1970), 1-11.
[2] V.V. Beresnevich, "On approximation of real numbers by real algebraic numbers", Acta Arith., 90:2 (1999), 97-112.
[3] V.V. Beresnevich, V.I. Bernik, D.Y. Kleinbock, G.A. Margulis, "Metric Diophantine approximation: the Khintchine-Groshev theorem for nondegenerate manifolds", Mosc. Math. J., 2:2 (2002), 203-225.
[4] V. Bernik, N. Budarina, D. Dickinson, "A divergent Khintchine theorem in the real, complex, and p-adic fields", Lith. Math. J., 48:2 (2008), 158-173.
[5] V.V Beresnevich, V.I. Bernik and E.I. Kovalevskaya, "On approximation of p-adic numbers by p-adic algebraic numbers", J. Number Theory., 111:1 (2005), 33-56.
[6] A. Baker, "Dirichlet theorem on Diophantine approximation", Math. Proc. Cam. Phil. Soc., 83:1 (1978), 37-59.
[7] V.I. Bernik, I.L. Morotskaya, "Diophantine approximation in Qp and Hausdorff dimension", Vestsi Akad. Navuk BSSR Ser. Fiz-Mat. nauk., 3 (1986), 3-9, 123.
[8] I.R. Dombrovsky, "Simultaneous approximation of real numbers by algebraic numbers of bounded degree", Dokl. Akad. Nauk BSSR, 3 (1989), 205-208.
[9] Y.V. Melnichuk, "Diophantine approximation on curves and Hausdorff dimension", Mat. Zametki, 26 (1979), 347-354.
[10] D.V. Vasilyev, "Diophantine sets in C and Hausdorff dimension", Inst. Math., Belarus. Acad. Sc., 1997, 21-28.
[11] Y. Bugeaud, Approximation by algebraic numbers, Cambridge Tracts in Mathematics, 160, Cambridge University Press, Cambridge, 2004.
[12] V.V. Beresnevich, "Effective estimates for measures of sets of real numbers with a given order of approximation by quadratic irrationalities", Vestsi Akad. Navuk Belarusi Ser. Fiz.-Mat. Navuk., 4 (1996), 10-15.
[13] V. Bernik, F. Gotze, O. Kukso, "Regular systems of real algebraic numbers of third degree in small intervals", Anal. Probab. Methods Number Theory, ed. A. Laurinchikas et., 2012, 61-68.
[14] V. Bernik, F. Gotze, "Distribution of real algebraic numbers of arbitrary degree in short intervals", Izv. Math., 79:1 (2015), 18-39.
[15] V. Sprindzuk, Mahler's problem in the metric theory of numbers, Amer. Math. Soc., 25, Providence, RI, 1969.

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