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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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