Dimensions of Fractional Brownian Images

This paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential-theoretic methods are used to produce dimension bounds for images of sets under Hölder maps and certain stochastic processes. We apply this to compute the almost-sure value of the dimension of Borel sets under index-$$\alpha $$ α fractional Brownian motion in terms of dimension profiles defined using capacities. As a corollary, this establishes continuity of the profiles for Borel sets and allows us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Hölder images such as projections. The methods used propose a general strategy for related problems; dimensional information about a set may be learned from analysing particular fractional Brownian images of that set. To conclude, we obtain bounds on the Hausdorff dimension of exceptional sets, with respect to intermediate dimensions, in the setting of projections.

Gaussian process approximations for multicolor Pólya urn models

Motivated by mathematical tissue growth modelling, we consider the problem of approximating the dynamics of multicolor Pólya urn processes that start with large numbers of balls of different colors and run for a long time. Using strong approximation theorems for empirical and quantile processes, we establish Gaussian process approximations for the Pólya urn processes. The approximating processes are sums of a multivariate Brownian motion process and an independent linear drift with a random Gaussian coefficient. The dominating term between the two depends on the ratio of the number of time steps n to the initial number of balls N in the urn. We also establish an upper bound of the form $c(n^{-1/2}+N^{-1/2})$ for the maximum deviation over the class of convex Borel sets of the step-n urn composition distribution from the approximating normal law.

Decompositions and measures on countable Borel equivalence relations

We show that the uniform measure-theoretic ergodic decomposition of a countable Borel equivalence relation $(X, E)$ may be realized as the topological ergodic decomposition of a continuous action of a countable group $\Gamma \curvearrowright X$ generating E. We then apply this to the study of the cardinal algebra $\mathcal {K}(E)$ of equidecomposition types of Borel sets with respect to a compressible countable Borel equivalence relation $(X, E)$ . We also make some general observations regarding quotient topologies on topological ergodic decompositions, with an application to weak equivalence of measure-preserving actions.

Boolean Valued Representation of Random Sets and Markov Kernels with Application to Large Deviations

We establish a connection between random set theory and Boolean valued analysis by showing that random Borel sets, random Borel functions, and Markov kernels are respectively represented by Borel sets, Borel functions, and Borel probability measures in a Boolean valued model. This enables a Boolean valued transfer principle to obtain random set analogues of available theorems. As an application, we establish a Boolean valued transfer principle for large deviations theory, which allows for the systematic interpretation of results in large deviations theory as versions for Markov kernels. By means of this method, we prove versions of Varadhan and Bryc theorems, and a conditional version of Cramér theorem.

Equivalence of codes for countable sets of reals

A set $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ is universal for countable subsets of ${\mathbb {R}}$ if and only if for all $x \in {\mathbb {R}}$ , the section $U_x = \{y \in {\mathbb {R}} : U(x,y)\}$ is countable and for all countable sets $A \subseteq {\mathbb {R}}$ , there is an $x \in {\mathbb {R}}$ so that $U_x = A$ . Define the equivalence relation $E_U$ on ${\mathbb {R}}$ by $x_0 \ E_U \ x_1$ if and only if $U_{x_0} = U_{x_1}$ , which is the equivalence of codes for countable sets of reals according to U. The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $E_{U^*}$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $E_U$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $E_U$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $\mathbf {\Sigma }_3^1$ -generic absoluteness, then for all $U \subseteq {\mathbb {R}} \times {\mathbb {R}}$ that are $\mathbf {\Sigma }_1^1$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $E_U$ .

ON GILP’S GROUP-THEORETIC APPROACH TO FALCONER’S DISTANCE PROBLEM

In this paper, we follow and extend a group-theoretic method introduced by Greenleaf–Iosevich–Liu–Palsson (GILP) to study finite points configurations spanned by Borel sets in $\mathbb{R}^n,n\geq 2,n\in\mathbb{N}.$ We remove a technical continuity condition in a GILP’s theorem in [Revista Mat. Iberoamer31 (2015), 799–810]. This allows us to extend the Wolff–Erdogan dimension bound for distance sets to finite points configurations with k points for $k\in\{2,\dots,n+1\}$ forming a $(k-1)$ -simplex.

THE GAME OPERATOR ACTING ON WADGE CLASSES OF BOREL SETS

We study the behavior of the game operator $$ on Wadge classes of Borel sets. In particular we prove that the classical Moschovakis results still hold in this setting. We also characterize Wadge classes ${\bf{\Gamma }}$ for which the class has the substitution property. An effective variation of these results shows that for all $1 \le \eta < \omega _1^{{\rm{CK}}}$ and $2 \le \xi < \omega _1^{{\rm{CK}}}$, is a Spector class while is not.