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On the rate of convergence to zero of the measure of extremal sets in metric theory of transcendental numbers. (English) Zbl 1443.11145
Let \(I\) be a measurable subset of \([0,1]\) (the Lebesgue measure \(\mu\) are used), \(Q>1\) be a positive integer. The following class of polynomials is considered: \[ \mathcal P_n(Q)=\{P\in \mathcal P_n: H(P)\le Q\}, \] where \(H=H(P)\) is the height of any polynomial \[ P=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0\in\mathcal P_n, \] i.e., the height is the maximum of the absolute values of coefficients of this polynomial. Also, for \(w>0\) and \(n< w\), the sets \(L_n(Q,w)=L_{n,I}(Q,w)\) of \(x\in I\) for which the inequality \[ |P(x)|< Q^{-w} \] has a solution in polynomials \(P\in \mathcal P_n(Q)\), are considered.
In the present article, the main attention is given to the question on the rate of convergence to zero of the measure of such set. In particular, the author notes that in this paper, for the first time, an effective estimate was obtained for this rate of convergence to zero.

MSC:
11J83 Metric theory
11K60 Diophantine approximation in probabilistic number theory
11J68 Approximation to algebraic numbers
11J13 Simultaneous homogeneous approximation, linear forms
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