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

Acta Arithmetica (Impact Factor: 0.42). 03/2008; DOI: 10.4064/aa117-1-4
Source: arXiv


This paper goes back to a famous problem of Mahler in metrical Diophantine approximation. The problem has been settled by Sprindzuk and subsequently improved by Alan Baker and Vasili Bernik. In particular, Bernik's result establishes a convergence Khintchine type theorem for Diophantine approximation by polynomials, that is it allows arbitrary monotonic error of approximation. In the present paper the monotonicity assumption is completely removed.

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Available from: Victor Beresnevich, May 27, 2015
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    • "In [2] it has been shown that if the sum in (21) diverges and ψ is monotonic, then for almost every real x inequality (20) holds infinitely often. More recently [3], the monotonicity assumption in Bernik's convergence result has been removed. However, removing the monotonicity assumption from the divergence result remains an open problem akin to the Duffin-Schaeffer conjecture. "
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    ABSTRACT: In this paper we discuss a general problem on metrical Diophantine approximation associated with a system of linear forms. The main result is a zero-one law that extends one-dimensional results of Cassels and Gallagher. The paper contains a discussion on possible generalisations including a selection of various open problems.
    Acta Arithmetica 04/2008; 133(4). DOI:10.4064/aa133-4-5 · 0.42 Impact Factor
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    • "Let P n,ψ be the set of all real numbers x such that |p(x)| < H(p)ψ(H(p)) (4·2) holds for infinitely many p ∈ Z n [X] with H(p) h 0 . V. Bernik [9] and V. Beresnevich [5] showed that P n,ψ has Lebesgue measure zero if h ψ(h)h n < ∞. In the opposite case, V. Beresnevich [3] established that P n,ψ has full measure in R. To our knowledge, the size properties of P n,ψ have not been studied any further. "
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