Ville Suomala

Existence of doubling measures via generalised nested cubes

Working on doubling metric spaces, we construct generalised dyadic cubes adapting ultrametric structure. If the space is complete, then the existence of such cubes and the mass distribution principle lead into a simple proof for the existence of doubling measures. As an application, we show that for each

Conical upper density theorems and porosity of measures

\epsilon>0

Nonsymmetric conical upper density and

there is a doubling measure having full measure on a set of packing dimension at most

k

\epsilon

-porosity

.

Packing dimension of mean porous measures

Abstract We study how measures with finite lower density are distributed around ( n − m ) -planes in small balls in R n . We also discuss relations between conical upper density theorems and porosity. Our results may be applied to a large collection of Hausdorff and packing type measures.

UPPER CONICAL DENSITY RESULTS FOR GENERAL MEASURES ON R-n

We study how the Hausdorff measure is distributed in nonsymmetric narrow cones in ℝ n . As an application, we find an upper bound close to n — k for the Hausdorff dimension of sets with large k-porosity. With k-porous sets we mean sets which have holes in k different directions on every small scale.

Local homogeneity and dimensions of measures

We prove that the packing dimension of any mean porous Radon measure on Rd may be estimated from above by a function which depends on mean porosity. The upper bound tends to d . 1 as mean porosity tends to its maximum value. This result was stated in D. B. Beliaev and S. K. Smirnov [�eOn dimension of porous measures�f, Math. Ann. 323 (2002) 123.141], and in a weaker form in E. J�Narvenp�Na�Na and M. J�Narvenp�Na�Na [�ePorous measures on Rn: local structure and dimensional properties�f, Proc. Amer. Math. Soc. (2) 130 (2002) 419.426], but the proofs are not correct. Quite surprisingly, it turns out that mean porous measures are not necessarily approximable by mean porous sets. We verify this by constructing an example of a mean porous measure �E on R such that �E(A) = 0 for all mean porous sets A �¼ R.

Hausdorff dimension of affine random covering sets in torus

We study conical density properties of general Borel measures on Euclidean spaces. Our results are analogous to the previously known result on the upper density properties of Hausdorff and packing-type measures.

Dimensions of random covering sets in Riemann manifolds

We introduce two new concepts, local homogeneity and local L^q-spectrum, both of which are tools that can be used in studying the local structure of measures. The main emphasis is given to the examination of local dimensions of measures in doubling metric spaces. As an application, we reach a new level of generality and obtain new estimates in the study of conical densities and porous measures.