Measure Theoretic Laws for Lim Sup Sets

By Victor Beresnevich, Detta Dickinson, Sanju Velani. 2004

248 citations recorded by the source corpus.

Given a compact metric space (X,d) equipped with a non-atomic, probability measure m and a real, positive decreasing function p we consider a `natural' class of limsup subsets La(p) of X. The classical limsup sets of `well approximable' numbers in the theory of metric Diophantine approximation fall within this class. We show that m(La(p))>0 under a `global ubiquity' hypothesis and the divergence of a certain m--volume sum. In fact, under a `local ubiquity' hypothesis we show that La(p) has full measure; i.e. m(La(p)) =1 . This is the analogue of the divergent part of the classical Khintchine-Groshev theorem in number theory. Moreover, if the 'local ubiquity' hypothesis is satisfied and a certain f-volume sum diverges then we are able to show that the Hausdorff f--measure of La(p) is infinite. A simple consequence of this is a lower bound for the Hausdorff dimension of La(p) and various results concerning the dimension and measure of related `exact order' sets. Essentially, the notion of `local ubiquity' unexpectedly unifies `divergent' type results for La(p) with respect to the natural measure m and general Hausdorff measures. Applications of the general framework include those from number theory, Kleinian groups and rational maps. Even for the classical limsup sets of `well approximable' numbers, the framework strengthens the classical Hausdorff measure result of Jarnik and opens up the Duffin-Schaeffer conjecture for Hausdorff measures.

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

View Paper: https://www.semanticscholar.org/paper/e3d3f9333d6640d9cfde4232520af78ab3468de6

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Citations in the corpus

Classical Metric Diophantine Approximation Revisited: The Khintchine–Groshev Theorem (2008, background): “This settles a multi-dimensional analogue of Catlin’s Conjecture.”
Hausdorff measure of sets of Dirichlet non-improvable numbers (2017, background): “assume that Ψ(t) ≥ 1 for all t≫ 1. Otherwise, Ψ(t) <1 for all large t since we have assumed Ψ to be non-decreasing. Then it is obvious that G1(Ψ), and thus G(Ψ), contains all irrational numbers in [0,1], and that the sum in Theorem 1.8 diverges. It is well known that G1(Ψ) contains K(3Ψ). The pro”
The shrinking target problem in the dynamical system of continued fractions (2014, methodology): “For this special case, the dimension of S({zn}n 1, f) follows from the theory developed by Beresnevich, Dickinson and Velani on the measure theoretical laws of limsup sets [1] (see also [2]).”
Rational points near manifolds and metric Diophantine approximation (2009, background): “The notion of ubiquity introduced below is equivalent to that of [7] in the setting that is now to be described.”
A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures (2004, background): “For numerous examples of lim sup sets and associated Khintchine type theorems (the approximating function ψ is assumed to be monotonic) within the framework of this section, the reader is referred to [1].”
Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces (2013, background): “Beginning with Patterson, many authors [5, 22, 39, 57, 67, 68, 71, 78, 79, 80] have considered generalizations of these theorems to the case of a nonelementary geometrically finite group acting on standard hyperbolic space X = H for some d ∈ N, considering a parabolic fixed point if G has at least o”
Heterogeneous ubiquitous systems in ℝd and Hausdorff dimension (2007, background): “• when the sequence {(xn, λn)}n forms an ubiquitous system [22, 23, 33, 15] with respect to a monofractal measure carried by the set S.”
Mass transference principle from rectangles to rectangles in Diophantine approximation (2019, background): “les. In this section, we introducethe notion called“local ubiquity for rectangles” which is the extended one of the notion “local ubiquity for balls” introduced by Beresnevich, Dickinson & Velani [7] and try to catch the nature of the rectangles inspired by Minkowski’s theorem. Fix an integer d≥”
A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation (2013, methodology): “A mass transference principle (MTP) was developed by Beresnevich and Velani [5] in order to study the Duffin– Schaeffer problem in the metric number theory (see also [4]).”
Diophantine approximation and badly approximable sets (2004, background): “To a certain extent, this paper complements [3] in which a general framework for establishing measure theoretic laws for ‘well approximable’ sets is established.”