arXiv subjects
Publications and source records attributed to Ville Suomala.
We consider sets in uniformly perfect metric spaces which are null for every doubling measure of the space or which have positive measure for all doubling measures. These sets are called thin and fat, respectively. In our main results, we give sufficient conditions for certain cut-out sets being thin or fat.
For a large class of Cantor sets on the real-line, we find sufficient and necessary conditions implying that a set has positive (resp. null) measure for all doubling measures of the real-line. We also discuss same type of questions for atomic doubling measures defined on certain midpoint Cantor sets.
We study the decay of $μ(B(x,r)\cap C)/μ(B(x,r))$ as $r\downarrow 0$ for different kinds of measures $μ$ on $\R^n$ and various cones $C$ around $x$. As an application, we provide sufficient conditions implying that the local dimensions can be calculated via cones almost everywhere.
We consider metrics on Euclidean domains $Ω\subset\R^n$ that are induced by continuous densities $ρ\colonΩ\rightarrow(0,\infty)$ and study the Hausdorff and packing dimensions of the boundary of $Ω$ with respect to these metrics.
We calculate the almost sure Hausdorff dimension for a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous Markov constructions.
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 $ε>0$ there is a doubling measure having full measure on a set of packing dimension at most $ε$.