Related papers: Cardinal invariants associated with Hausdorff meas…
Davies and Rogers constructed a Hausdorff measure satisfying the following property: every Borel subset of the space has measure either $\infty$ or $0$. In this paper, we examine cardinal invariants of their measure.
It is shown to be consistent with set theory that the uniformity invariant for Lebesgue measure is strictly greater than the corresponding invariant for Hausdorff r-dimensional measure where 0<r<1.
J. Zapletal asked if all the forcing notions considered in his monograph are homogeneous. Specifically, he asked if the forcing consisting of Borel sets of $\sigma$-finite 2-dimensional Hausdorff measure in $\mathbb{R}^3$ (ordered under…
We introduce a two-parameter modification of the cofinality invariant of ideals. This allows us to include the interaction of a pair of ideals in the study of base-like structures. We find the values (cardinal numbers or well-known cardinal…
Using a countable support product of creature forcing posets, we show that consistently, for uncountably many different functions the associated Yorioka ideals' uniformity numbers can be pairwise different. In addition we show that, in the…
Critical circle homeomorphisms have an invariant measure totally singular with respect to the Lebesgue measure. We prove that singularities of the invariant measure are of Holder type. The Hausdorff dimension of the invariant measure is…
Yorioka [J. Symbolic Logic 67(4):1373-1384, 2002] introduced a class of ideals (parametrized by reals) on the Cantor space to prove that the relation between the size of the continuum and the cofinality of the strong measure zero ideal on…
In [CMRM24], it was proved that it is relatively consistent that \emph{bounding number} $\mathfrak{b}$ is smaller than the uniformity of $\mathcal{MA}$, where $\mathcal{MA}$ denotes the ideal of the meager-additive sets of $2^{\omega}$. To…
This paper provides an extensive study of the $\mathscr{I}$-Miller null ideals $M_\mathscr{I}$, $\sigma$-ideals on the Baire space parametrized by ideals $\mathscr{I}$ on countable sets. These $\sigma$-ideals are associated to the idealized…
This is a revision (and partial retraction) of my previous abstarct. Let $\lambda(X)$ denote Lebesgue measure. If $X\subseteq [0,1]$ and $r \in (0,1)$ then the $r$-Hausdorff capacity of $X$ is denoted by $H^r(X)$ and is defined to be the…
We prove two $\mathrm{ZFC}$ inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of $\omega$ of asymptotic density $0$. We obtain…
Inspired by Bartoszy\'nski's work on small sets, we introduce a new ideal defined by interval partitions on natural numbers and summable sequences of positive reals. Similarly, we present another ideal that relies on Bartoszy\'nski's and…
We classify the measures on SL (k,R)/SL (k,Z) which are invariant and ergodic under the action of the group A of positive diagonal matrices with positive entropy. We apply this to prove that the set of exceptions to Littlewood's conjecture…
We consider subsets of the (symbolic) sequence space that are invariant under the action of the semigroup of multiplicative integers. A representative example is the collection of all 0-1 sequences $(x_k)$ such that $x_k x_{2k}=0$ for all…
Much recent work in cardinal characteristics has focused on generalizing results about $\omega$ to uncountable cardinals by studying analogues of classical cardinal characteristics on the generalized Baire and Cantor spaces $\kappa^\kappa$…
The main result of this paper is an improvement of the upper bound on the cardinal invariant ${\mathord{\mathrm{cov}}}^{\ast}({\mathcal{Z}}_{0})$ that was discovered by Raghavan and Shelah in an earlier paper. Here ${\mathcal{Z}}_{0}$ is…
We give explicit bounds for the Hausdorff dimension of the unique invariant measure of $C^3$ multicritical circle maps without periodic points. These bounds depend only on the arithmetic properties of the rotation number.
We study the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding nonconformal map on the torus given by an integer-valued diagonal matrix. The Hausdorff dimension of a "general Sierpinski…
The paper is an extensive and systematic study of cardinal invariants we call slalom numbers, describing the combinatorics of sequences of sets of natural numbers. Our general approach, based on relational systems, covers many such cardinal…
We study various measure theories using the classical approach and then compute the Hausdorff dimension of some simple objects and self-similar fractals. We then develop a nonstandard approach to these measure theories and examine the…