zbMATH — the first resource for mathematics

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.

11J83 Metric theory
11K60 Diophantine approximation in probabilistic number theory
11J68 Approximation to algebraic numbers
11J13 Simultaneous homogeneous approximation, linear forms
Full Text: DOI
[1] Beresnevich, VV, On approximation of real numbers by real algebraic numbers, Acta Arith., 90, 97-112, (1999) · Zbl 0937.11027
[2] Beresnevich, VV, A Groshev type theorem for convergence on manifolds, Acta Math. Hung., 94, 99-130, (2002) · Zbl 0997.11053
[3] Bernik, VI; Kleinbok, D.; Margulis, G., Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions, Int. Math. Not., 2001, 453-486, (2001) · Zbl 0986.11053
[4] Beresnevich, VV; Bernik, VI; Kleinbok, D.; Margulis, G., Metric Diophantine approximation: the Khintchine-Groshev theorem for nondegenerate manifolds, Mosc. Math. J., 2, 203-225, (2002) · Zbl 1013.11039
[5] Bernik, V., The metric theorem on the simultaneous approximation of zero by values of integral polynomials, Izv. Akad. Nauk SSSR Ser. Mat., 44, 24-45, (1980) · Zbl 0426.10055
[6] Bernik, V., An application of Hausdorff dimension in the theory of Diophantine Approximation, Acta Arith., 42, 219-253, (1983) · Zbl 0482.10049
[7] Bernik, V., On the exact order of approximation of zero by values of integral polynomials, Acta Arith., 53, 17-28, (1989) · Zbl 0692.10042
[8] Bernik, V.; Budarina, N.; Dickinson, D., A divergent Khintchine theorem in the real, complex and \(p\)-adic fields, Lith. Math. J., 48, 158-173, (2008) · Zbl 1143.11337
[9] Bernik, V.; Budarina, N.; Dickinson, D., Simultaneous Diophantine approximation in the real, complex, and \(p\)-adic fields, Math. Proc. Camb. Philos. Soc., 149, 193-216, (2010) · Zbl 1221.11159
[10] Bugeaud, Y.: Approximation by Algebraic Numbers. Cambridge Tracts in Mathematics, vol. 160. Cambridge University Press, Cambridge (2004) · Zbl 1055.11002
[11] Khintchine, AJ, Zwei Bemerkungen zu einer Arbeit des Herrn Perron, Math. Z., 22, 274-284, (1925) · JFM 51.0157.02
[12] Sprindzuk, V.G.: Mahler’s problem in metric number theory. Nauka i Tehnika Minsk (1967) [Transl. Math. Monogr., vol. 25, Am. Math. Soc., Providence, R.I. (1969)]
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.