Article

# On a theorem of V. Bernik in the metrical theory of Diophantine approximation

Acta Arithmetica (Impact Factor: 0.47). 03/2008; DOI: 10.4064/aa117-1-4

Source: arXiv

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**ABSTRACT:**A central problem motivated by Diophantine approximation is to determine the size properties of subsets of of the formMathematical Proceedings of the Cambridge Philosophical Society 12/2007; 144(01):119 - 144. · 0.68 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In the last twenty years, the exact order of approximation by zeros of the moduli of the values of the integer polynomials in a real and a complex variable was established. However, in the case of convergence of the series consisting of the right-hand sides of inequalities, the monotonicity condition for the right-hand sides in the classical Khintchine theorem can be dropped. It is shown in the present paper that, in the complex case, the monotonicity condition is also insignificant for polynomials of arbitrary degree.Mathematical Notes 93(5-6). · 0.24 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we show that if the volume sum $ \sum\nolimits_{h = 1}^\infty {{h^{n - 1}}{\Psi^t}(h)} $ converges for a function ψ (not necessarily monotonic), then the set of points $ \left( {x,{w_1}, \ldots, {w_{t - 1}}} \right) \in {\mathbb R} \times {{\mathbb Q}_{{p_1}}} \times \ldots \times {{\mathbb Q}_{{p_{t - 1}}}} $ that simultaneously satisfy the inequalities $ \left| {P(x)} \right| < \Psi (H) {\text{and}} {\left| {P\left( {{w_i}} \right)} \right|_{{p_i}}} < \Phi (H), 1 \leqslant i \leqslant t - 1 $ , for infinitely many integer polynomials P has measure zero.Lithuanian Mathematical Journal 01/2011; 51(4). · 0.35 Impact Factor

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