an:06198923
Zbl 1338.11064
Budarina, N. V.
The Mahler problem with nonmonotone right-hand side in the field of complex numbers
EN
Math. Notes 93, No. 6, 802-809 (2013); translation from Mat. Zametki 93, No. 6, 812-820 (2013).
00322383
2013
j
11J83 13P05
integer polynomials; classical Khintchine theorem; Lebesgue measure; Baker's conjecture
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 \textit{V. Beresnevich} [Acta Arith. 117, No. 1, 71--80 (2005; Zbl 1201.11078)].
Mumtaz Hussain (Callaghan)
Zbl 1201.11078