55 research outputs found
Spatially independent martingales, intersections, and applications
We define a class of random measures, spatially independent martingales,
which we view as a natural generalisation of the canonical random discrete set,
and which includes as special cases many variants of fractal percolation and
Poissonian cut-outs. We pair the random measures with deterministic families of
parametrised measures , and show that under some natural
checkable conditions, a.s. the total measure of the intersections is H\"older
continuous as a function of . This continuity phenomenon turns out to
underpin a large amount of geometric information about these measures, allowing
us to unify and substantially generalize a large number of existing results on
the geometry of random Cantor sets and measures, as well as obtaining many new
ones. Among other things, for large classes of random fractals we establish (a)
very strong versions of the Marstrand-Mattila projection and slicing results,
as well as dimension conservation, (b) slicing results with respect to
algebraic curves and self-similar sets, (c) smoothness of convolutions of
measures, including self-convolutions, and nonempty interior for sumsets, (d)
rapid Fourier decay. Among other applications, we obtain an answer to a
question of I. {\L}aba in connection to the restriction problem for fractal
measures.Comment: 96 pages, 5 figures. v4: The definition of the metric changed in
Section 8. Polishing notation and other small changes. All main results
unchange
A note on the hitting probabilities of random covering sets
Let be the random covering set on
the torus , where is a sequence of ball-like sets and
is a sequence of independent random variables uniformly distributed on
\T^d. We prove that almost surely whenever
is an analytic set with Hausdorff dimension,
, where is the almost sure Hausdorff dimension of
. Moreover, examples are given to show that the condition on
cannot be replaced by the packing dimension of .Comment: 11 page
Dimension of the boundary in different metrics
We consider metrics on Euclidean domains that are induced
by continuous densities and study the
Hausdorff and packing dimensions of the boundary of with respect to
these metrics.Comment: 20 pages, 2 figure
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
there is a doubling measure having full measure on a set of
packing dimension at most
Dimension, entropy, and the local distribution of measures
We present a general approach to the study of the local distribution of
measures on Euclidean spaces, based on local entropy averages. As concrete
applications, we unify, generalize, and simplify a number of recent results on
local homogeneity, porosity and conical densities of measures.Comment: v2: 23 pages, 6 figures. Updated references. Accepted to J. London
Math. So
Thin and fat sets for doubling measures in metric spaces
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
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