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The Mahler problem with nonmonotone right-hand side in the field of complex numbers. (English. Russian original) Zbl 1338.11064
Math. Notes 93, No. 6, 802-809 (2013); translation from Mat. Zametki 93, No. 6, 812-820 (2013).
The paper under review deals with a problem in metric Diophantine approximation. To state the result of this paper we first introduce some notation. Let $$P_n$$ denote the set of integer polynomials of degree at most $$n$$ and let $$\Psi$$ be a positive function. Let $$W$$ be the set of all complex numbers $$z$$ which satisfy the inequality $$|P(z)|<\Psi(H(P))$$ for infinitely many integer polynomials $$P\in P_n$$, where $$H(P)$$ is the maximum of the modulus of the integer coefficients of $$P$$. It is proved that, for $$n\geq 3$$ the Lebesgue measure of the set $$W$$ is zero if $\sum_{k=1}^\infty k^{n-2}\Psi^2(k)<\infty.$ The main novelty of this result is that the function $$\Psi$$ is non-monotonic. The same result over the set of real numbers was established by V. Beresnevich [Acta Arith. 117, No. 1, 71–80 (2005; Zbl 1201.11078)].

MSC:
 11J83 Metric theory 13P05 Polynomials, factorization in commutative rings
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References:
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