Abstract
In the last twenty years, the exact order of approximation by zeros of the moduli of the values of the integer polynomials in a real and a complex variable was established. However, in the case of convergence of the series consisting of the right-hand sides of inequalities, the monotonicity condition for the right-hand sides in the classical Khintchine theorem can be dropped. It is shown in the present paper that, in the complex case, the monotonicity condition is also insignificant for polynomials of arbitrary degree.
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References
K. Mahler, “Über das Maß der Menge aller S-Zahlen,” Math. Ann. 106(1), 131–139 (1932).
V. G. Sprindzhuk, “On a conjecture of Mahler,” Dokl. Akad. Nauk SSSR 154(4), 783–786 (1964) [Soviet Math. Dokl. 5, 183–187 (1964)].
V. G. Sprindzhuk, The Mahler Problem in the Metric Theory of Numbers (Nauka i Tekhnika, Minsk, 1967) [in Russian].
V. I. Bernik and D. V. Vasil’ev, “Khintchine-type theorem for integral polynomials of a complex variable,” Tr. Inst. Mat. NAN Belarus 3, 10–20 (1999). [in Russian].
A. I. Khintchine, “Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen,” Math. Ann. 92(1–2), 115–125 (1924).
V. V. Beresnevich, “On a theorem of V. Bernik in the metric theory of Diophantine approximation,” Acta Arith. 117(1), 71–80 (2005).
V. I. Bernik, D. Kleinbock, and G. A. Margulis, “Khintchine-type theorems on manifolds: the convergence case for standart and multiplicative versions,” Int. Math. Res. Notices 9, 453–486 (2001).
V. I. Bernik, “The exact order of approximating zero by values of integral polynomials,” Acta Arith. 53(1), 17–28 (1989).
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Original Russian Text © N. V. Budarina, 2013, published in Matematicheskie Zametki, 2013, Vol. 93, No. 6, pp. 812–820.
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Budarina, N.V. The Mahler problem with nonmonotone right-hand side in the field of complex numbers. Math Notes 93, 802–809 (2013). https://doi.org/10.1134/S0001434613050192
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DOI: https://doi.org/10.1134/S0001434613050192