Journal of Mathematical Analysis and Applications, 2007
We prove that if a self-similar set E in R n with Hausdorff dimension s satisfies the strong sepa... more We prove that if a self-similar set E in R n with Hausdorff dimension s satisfies the strong separation condition, then the maximal values of the H s -density on the class of arbitrary subsets of R n and on the class of Euclidean balls are attained, and the inverses of these values give the exact values of the Hausdorff and spherical Hausdorff measure of E. We also show that a ball of minimal density exists, and the inverse density of this ball gives the exact packing measure of E. Lastly, we show that these elements of optimal densities allow us to construct an optimal almost covering of E by arbitrary subsets of R n , an optimal almost covering of E by balls and an optimal packing of E.
We analyze self-similarity with respect to infinite sets of similitudes from a measure-theoretic ... more We analyze self-similarity with respect to infinite sets of similitudes from a measure-theoretic point of view. We extend classic results for finite systems of similitudes satisfying the open set condition to the infinite case. We adopt Vitali-type techniques to approximate overlapping self-similar sets by non-overlapping self-similar sets. As an application we show that any open and bounded set $A \subseteq \mathbb{R}^n$ with a boundary of null Lebesgue measure always contains a self-similar set generated by a countable system of similitudes and with Lebesgue measure equal to that ofA.
Journal of Mathematical Analysis and Applications, 2007
We prove that if a self-similar set E in R n with Hausdorff dimension s satisfies the strong sepa... more We prove that if a self-similar set E in R n with Hausdorff dimension s satisfies the strong separation condition, then the maximal values of the H s -density on the class of arbitrary subsets of R n and on the class of Euclidean balls are attained, and the inverses of these values give the exact values of the Hausdorff and spherical Hausdorff measure of E. We also show that a ball of minimal density exists, and the inverse density of this ball gives the exact packing measure of E. Lastly, we show that these elements of optimal densities allow us to construct an optimal almost covering of E by arbitrary subsets of R n , an optimal almost covering of E by balls and an optimal packing of E.
We analyze self-similarity with respect to infinite sets of similitudes from a measure-theoretic ... more We analyze self-similarity with respect to infinite sets of similitudes from a measure-theoretic point of view. We extend classic results for finite systems of similitudes satisfying the open set condition to the infinite case. We adopt Vitali-type techniques to approximate overlapping self-similar sets by non-overlapping self-similar sets. As an application we show that any open and bounded set $A \subseteq \mathbb{R}^n$ with a boundary of null Lebesgue measure always contains a self-similar set generated by a countable system of similitudes and with Lebesgue measure equal to that ofA.
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Papers by Manuel Moran