in this work are studied the jacobians of certain singular transformations and the corresponding measures which support the jacobian computations.

The Falconer conjecture says that if a compact planar set has Hausdorff dimension > 1, then the Euclidean distance set ∆(E) = {|x − y| : x, y ∈ E} has positive Lebesgue measure. In this paper we prove, under the same assumptions, that for almost every ellipse K, ∆ K (E) = {||x − y|| K : x, y ∈ E} has positive Lebesgue measure, where || · || K is the norm induced by an ellipse K. Equivalently, w...

These are some brief notes on measure theory, concentrating on Lebesgue measure on Rn. Some missing topics I would have liked to have included had time permitted are: the change of variable formula for the Lebesgue integral on Rn; absolutely continuous functions and functions of bounded variation of a single variable and their connection with Lebesgue-Stieltjes measures on R; Radon measures on ...

The concern of this paper is with effective packing dimension. This concept can be traced back to the work of Borel and Lebesgue who studied measure as a way of specifying the size of sets. Carathéodory later generalized Lebesgue measure to the n-dimensional Euclidean space, and this was taken further by Hausdorff [Hau19] who generalized the notion of s-dimensional measure to include non-intege...

Let λ(X) denote Lebesgue measure. If X ⊆ [0, 1] and r ∈ (0, 1) then the r-Hausdorff capacity of X is denoted by H(X) and is defined to be the infimum of all ∑ ∞ i=0 λ(Ii) r where {Ii}i∈ω is a cover of X by intervals. The r Hausdorff capacity has the same null sets as the r-Hausdorff measure which is familiar from the theory of fractal dimension. It is shown that, given r < 1, it is possible to ...

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 ...

In this work are studied the Jacobians of certain singular transformations and the corresponding measures which support the jacobian computations.