On the Hausdorff dimension of certain sets arising in Number Theory
نویسنده
چکیده
Any real number x in the unit interval can be expressed as a continued fraction x = [n1, ..., nN , ...]. Subsets of zero measure are obtained by imposing simple conditions on the n N . By imposing n N ≤ m ∀ N ∈ IN , Jarnik defined the corresponding sets Em and gave a first estimate of dH(Em), dH the Hausdorff dimension. Subsequent authors improved these estimates. In this paper we deal with dH(Em) and dH(Fm), Fm being the set of real numbers for which ∑N i=1 ni N ≤ m.
منابع مشابه
Hausdorff dimension of sets of divergence arising from continued fractions
A complex continued fraction can be represented by a sequence of Möbius transformations in such a way that the continued fraction converges if and only if the sequence converges at the origin. The set of divergence of the sequence of Möbius transformations is equivalent to the conical limit set from Kleinian group theory, and it is closely related to the Julia set from complex dynamics. We dete...
متن کاملMeasure theoretic laws for lim–sup sets
Given a compact metric space (Ω, d) equipped with a non-atomic, probability measure m and a real, positive decreasing function ψ we consider a ‘natural’ class of lim sup subsets Λ(ψ) of Ω. The classical lim sup sets of ‘well approximable’ numbers in the theory of metric Diophantine approximation fall within this class. We show that m(Λ(ψ)) > 0 under a ‘global ubiquity’ hypothesis and the diverg...
متن کاملExplicit Bounds for the Hausdorff Dimension of Certain Self-Affine Sets
A lower bound of the Hausdorff dimension of certain self-affine sets is given. Moreover, this and other known bounds such as the box dimension are expressed in terms of solutions of simple equations involving the singular values of the affinities. Keyword Codes: G.2.1;G.3
متن کاملTheory of dimension for large discrete sets and applications
We define two notions of discrete dimension based on the Minkowski and Hausdorff dimensions in the continuous setting. After proving some basic results illustrating these definitions, we apply this machinery to the study of connections between the Erdős and Falconer distance problems in geometric combinatorics and geometric measure theory, respectively.
متن کاملHausdorff Dimension of Limit Sets for Parabolic Ifs with Overlaps
We study parabolic iterated function systems with overlaps on the real line. We show that if a d-parameter family of such systems satisfies a transversality condition, then for almost every parameter value the Hausdorff dimension of the limit set is the minimum of 1 and the least zero of the pressure function. Moreover, the local dimension of the exceptional set of parameters is estimated. If t...
متن کامل