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...

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.

We establish formulas for bounds on the Haudorff measure of the intersection of certain Cantor sets with their translates. As a consequence we obtain a formula for the Hausdorff dimensions of these intersections.

In this article, we present some fixed point and coupled theorems adapted from the notion of F-contraction mappings in Banach spaces (B.S.) via measure noncompactness (M.N.C). Then define a new class generalized F-contractions, to upgrade results Falest Latrach (Bull Bell Math Soc Simon Stevin 22:797–812, 2015). Furthermore, investigate solvability system Volterra integral equations Algebra. Fi...