Classical Metric Diophantine Approximation Revisited: The Khintchine–Groshev Theorem

By Victor Beresnevich, Sanju Velani. 2008

78 citations recorded by the source corpus.

Let denote the set of ψ-approximable points in . Under the assumption that the approximating function ψ is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of . The famous Duffin-Schaeffer counterexample shows that the monotonicity assumption on ψ is absolutely necessary when m = n = 1. On the other hand, it is known that monotonicity is not necessary when n ≥ 3 (Schmidt) or when n = 1 and m ≥ 2 (Gallagher). Surprisingly, when n = 2, the situation is unresolved. We deal with this remaining case and thereby remove all unnecessary conditions from the classical Khintchine-Groshev theorem. This settles a multidimensional analog of Catlin's conjecture.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2cc5eea2a63e8532a0f14027450a0e866948810c

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Citations in the corpus

Metric Diophantine Approximation : aspects of recent work (2016, background): “The m = 2 situation was resolved only recently in [26], where it was shown that the monotonicity condition can be safely removed.”