For any , let be its dyadic expansion. Call , the -th maximal run-length function of . P. Erdös and A. Rényi showed that almost surely. This paper is concentrated on the points violating the above law. The size of sets of points, whose run-length function assumes on other possible asymptotic behaviors than , is quantified by their Hausdorff dimension.
This is a survey on transformation of fractal type sets and measures under orthogonal projections and more general mappings.
Let F : U ⊂ Rn → Rm be a differentiable function and p < m an integer. If k ≥ 1 is an integer, α ∈ [0, 1] and F ∈ Ck+(α), if we set Cp(F) = {x ∈ U | rank(Df(x)) ≤ p} then the Hausdorff measure of dimension (p + (n-p)/(k+α)) of F(Cp(F)) is zero.
We investigate the following question: under which conditions is a σ-compact partial two point set contained in a two point set? We show that no reasonable measure or capacity (when applied to the set itself) can provide a sufficient condition for a compact partial two point set to be extendable to a two point set. On the other hand, we prove that under Martin's Axiom any σ-compact partial two point set such that its square has Hausdorff 1-measure zero is extendable.
We consider (bounded) Besicovitch sets in the Heisenberg group and prove that Lp estimates for the Kakeya maximal function imply lower bounds for their Heisenberg Hausdorff dimension.
The generalized Riemann integral of Pfeffer (1991) is defined on all bounded subsets of , but it is additive only with respect to pairs of disjoint sets whose closures intersect in a set of -finite Hausdorff measure of codimension one. Imposing a stronger regularity condition on partitions of sets, we define a Riemann-type integral which satisfies the usual additivity condition and extends the integral of Pfeffer. The new integral is lipeomorphism-invariant and closed with respect to the formation...
Let X be a complete metric space and write (X) for the family of all Borel probability measures on X. The local dimension of a measure μ ∈ (X) at a point x ∈ X is defined by whenever the limit exists, and plays a fundamental role in multifractal analysis. It is known that if a measure μ ∈ (X) satisfies a few general conditions, then the local dimension of μ exists and is equal to a constant for μ-a.a. x ∈ X. In view of this, it is natural to expect that for a fixed x ∈ X, the local dimension...