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

(Impact Factor: 0.42). 03/2008; 117(1). DOI: 10.4064/aa117-1-4
Source: arXiv

ABSTRACT

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. "
##### Article: A note on zero-one laws in metrical Diophantine approximation
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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.
Full-text · Article · Apr 2008 · Acta Arithmetica
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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. "
##### Article: Sets with large intersection and ubiquity
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ABSTRACT: A central problem motivated by Diophantine approximation is to determine the size properties of subsets of of the form where denotes an arbitrary norm, I a denumerable set, (xi,ri)i I a family of elements of × (0, ∞) and a nonnegative nondecreasing function defined on [0, ∞). We show that if FId, where Id denotes the identity function, has full Lebesgue measure in a given nonempty open subset V of , the set F belongs to a class Gh(V) of sets with large intersection in V with respect to a given gauge function h. We establish that this class is closed under countable intersections and that each of its members has infinite Hausdorff g-measure for every gauge function g which increases faster than h near zero. In particular, this yields a sufficient condition on a gauge function g such that a given countable intersection of sets of the form F has infinite Hausdorff g-measure. In addition, we supply several applications of our results to Diophantine approximation. For any nonincreasing sequence ψ of positive real numbers converging to zero, we investigate the size and large intersection properties of the sets of all points that are ψ-approximable by rationals, by rationals with restricted numerator and denominator and by real algebraic numbers. This enables us to refine the analogs of Jarník's theorem for these sets. We also study the approximation of zero by values of integer polynomials and deduce several new results concerning Mahler's and Koksma's classifications of real transcendental numbers.
Full-text · Article · Dec 2007 · Mathematical Proceedings of the Cambridge Philosophical Society
• ##### Article: Distribution of algebraic numbers and metric theory of Diophantine approximation
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ABSTRACT: In this paper we give an overview of recent results regarding close conjugate algebraic numbers, the number of integral polynomials with small discriminant and pairs of polynomials with small resultants.
No preview · Article · Jan 2013