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Algebraic points on the plane. (English. Russian original) Zbl 1204.11120
J. Math. Sci., New York 146, No. 2, 5680-5685 (2007); translation from Fundam. Prikl. Mat. 11, No. 6, 73-80 (2005).
Summary: This article gives quantitative estimates for the measure of points \((x, y)\) of a given rectangle admitting the construction of polynomials \(P(t)\) with small (with respect to the height of the polynomial) values of \(P(x)\) and \(P(y)\). Such estimates can be used in the problem of distribution of algebraic points on the plane.
11J83 Metric theory
11J13 Simultaneous homogeneous approximation, linear forms
Full Text: DOI
[1] V. V. Beresnevich, ”On approximation of real numbers by real algebraic integers,” Acta Arith., 90, No. 2, 97–112 (1999). · Zbl 0937.11027
[2] V. V. Beresnevich, ”A Groshev type theorem for convergence on manifolds,” Acta Math. Acad. Sci. Hungar., 94, Nos. 1–2, 99–130 (2002). · Zbl 0997.11053
[3] V. V. Beresnevich, ”Distribution of rational points near parabola,” Dokl. Akad. Nauk Belarusi, 46, 13–15 (2002). · Zbl 1177.11057
[4] V. V. Beresnevich, V. Bernik, D. Kleinbock, and G. Margulis, ”Metric Diophantine approximation: The Khinchine-Groshev type theorem for nondegenerable manifolds,” Moscow Math. J., 2, No. 2, 203–225 (2000). · Zbl 1013.11039
[5] V. I. Bernik, ”A metric theorem on the simultaneous approximation of zero by the values of integral polynomials,” Math. USSR-Izv., 16, No. 1, 21–40 (1981). · Zbl 0464.10041
[6] V. I. Bernik, ”On the exact order of approximation of zero by values of integral polynomials,” Acta Arith., 53, No. 1, 17–28 (1989). · Zbl 0692.10042
[7] V. I. Bernik and V. N. Borbat, ”Simultaneous approximation of zero by values of integral polynomials,” Proc. Steklov Inst. Math., 218, 53–68 (1997). · Zbl 0917.11035
[8] V. Bernik, D. Kleinbock, and G. Margulis, ”Khinchine-type theorem on manifolds: The convergence case and multiplicative versions,” Intern. Math. Research Notes, No. 9, 453–486 (2001). · Zbl 0986.11053
[9] M. Huxley, Area, Lattice Points and Exponential Sums, Oxford (1996). · Zbl 0861.11002
[10] N. A. Pereverseva, ”Simultaneous approximation of zero by values of relatively prime integral polynomials,” Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. (1984).
[11] V. G. Sprindzuk, Mahler’s Problem in Metric Number Theory, Amer. Math. Soc., Providence (1969).
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