For K ⊂ C compact, we say that K has vanishing analytic capacity (or γ(K) = 0) when all bounded analytic functions on C\K are constant. We would like to characterize γ(K) = 0 geometrically. Easily, γ(K) > 0 when K has Hausdorff dimension larger than 1, and γ(K) = 0 when dim(K) < 1. Thus only the case when dim(K) = 1 is interesting. So far there is no characterization of γ(K) = 0 in general, but...

There are two fundamental results in the classical theory of metric Diophantine approximation: Khintchine’s theorem and Jarńık’s theorem. The former relates the size of the set of well approximable numbers, expressed in terms of Lebesgue measure, to the behavior of a certain volume sum. The latter is a Hausdorff measure version of the former. We start by discussing these theorems and show that ...

It turns out that H1 is a measure, now called one-dimensional Hausdorff measure because it was generalized by Hausdorff [10] to the whole family of measures Hα, where α is any positive number (integer or noninteger). The modern theory of “fractals” is largely based on the notion of the Hausdorff dimension dimH (F) of a set F , defined by dimH (F) = inf{α > 0 : Hα(F) = 0}. We recommend the book ...

We show that, given a set E ⊂ Rn+1 with finite n-Hausdorff measure Hn, if the n-dimensional Riesz transform

We construct α-Hölder continuous functions on the real line so that all their level sets have positive (1−α)-dimensional Hausdorff measure.

We consider the problem of the minimization of the p-compliance functional where the control variables Σ are taking among closed connected one-dimensional sets. we prove some estimate from below of the p-compliance functional in terms of the one-dimensional Hausdorff measure of Σ and compute the value of the constant θ(p) appearing usually in Γ-limit functional of the rescaled p-compliance func...

The compact operators on the Riesz sequence space 1 ∞ have been studied by Başarır and Kara, “IJST (2011) A4, 279-285”. In the present paper, we will characterize some classes of compact operators on the normed Riesz sequence spaces and by using the Hausdorff measure of noncompactness.

Let E ⊂ R with H(E) < ∞, where H stands for the n-dimensional Hausdorff measure. In this paper we prove that E is n-rectifiable if and only if the limit

In the present survey paper, we explain how the theory of Hausdorff dimension and Hausdorff measure is used to answer certain questions in Diophantine approximation. The final section is devoted to a discussion around the Diophantine properties of the points lying in the middle third Cantor set.