Geodesic
On metric estimates of approximation properties of monic polynomial values
Trudy Instituta matematiki, Tome 21 (2013) no. 1, pp. 109-122

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The cases of convergence are proved in several analogues of Khinchine's theorem for approximation of zero by values of monic integer polynomials, including the case of nonmonotonic functions in the right-hand side of the inequalities and simultaneous approximations in $\mathbb{R}^2$. 
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N. V. Shamukova; N. I. Kalosha. On metric estimates of approximation properties of monic polynomial values. Trudy Instituta matematiki, Tome 21 (2013) no. 1, pp. 109-122. http://geodesic.mathdoc.fr/item/TIMB_2013_21_1_a13/