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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…

Let $\psi:\mathbb{N}\rightarrow\mathbb{R}_+$ be a monotonically non-increasing function, and let $\psi_v:\mathbb{N}\rightarrow\mathbb{R}_+$ be defined by $\psi_v(q)=1/q^v$. In this article, we consider self-similar sets whose iterated…

We study H-structures associated to SU-rank 1 measurable structures. We prove that the SU-rank of the expansion is continuous and that it is uniformly definable in terms of the parameters of the formulas. We also introduce notions of…

Hausdorff relation, topologically identifying points in a given space, belongs to elementary tools of modern mathematics. We show that if subtle enough mathematical methods are used to analyze this relation, the conclusions may be…

The paper aims to generalize Clausen's identity to the square of any Gauss hypergeometric function. Accordingly, solutions of the related 3rd order linear differential equation are found in terms of certain bivariate series that can reduce…

We consider expanding maps such that the unit interval can be represented as a full symbolic shift space with bounded distortion. There are already theorems about the Hausdorff dimension for sets defined by the set of accumulation points…

A class of nets in constructive (in A.A.Markov's sense) topological space for which the convergence is equivalent to convergence of all subsequences, is described. B.A.Kushner's theorem about coincidence of strong and weak constructive…

In this note we derive an upper bound for the Hausdorff dimension of the stable set of a hyperbolic set $\Lambda$ of a $C^2$ diffeomorphisms on a $n$-dimensional manifold. As a consequence we obtain that $\dim_H W^s(\Lambda)=n$ is…

In this paper we show that the Hausdorff dimension of the set of singular pairs is 4/3. We also show that the action of diag(e^t,e^t,e^{-2t}) on SL(3,R)/SL(3,Z) admits divergent trajectories that exit to infinity at arbitrarily slow…

Classical $W$-algebras in higher dimensions are constructed. This is achieved by generalizing the classical Gel'fand-Dickey brackets to the commutative limit of the ring of classical pseudodifferential operators in arbitrary dimension.…

For each prime p and a monic polynomial f, invertible over p, we define a group G_{p,f} of p-adic automorphisms of the p-ary rooted tree. The groups are modeled after the first Grigorchuk group, which in this setting is the group…

In this paper we prove some lower bounds on the Hausdorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling dimension functions. With some additional…

Let $A$ be a limsup random fractal with indices $\gamma_1, ~\gamma_2 ~$and $\delta$ on $[0,1]^d$. We determine the hitting probability $\mathbb{P}(A\cap G)$ for any analytic set $G$ with the condition $(\star)$$\colon$ $\dim_{\rm…

We study the class of transversal submanifolds. We characterize their blow-ups at transversal points and prove a negligibility theorem for their "generalized characteristic set", with respect to the Carnot-Carath\'eodory Hausdorff measure.…

On the probability simplex, we can consider the standard information geometric structure with the e- and m-affine connections mutually dual with respect to the Fisher metric. The geometry naturally defines submanifolds simultaneously…

We study $\Sigma^1_2$ definable counterparts for some algebraic equivalent forms of the Continuum Hypothesis. All turn out to be equivalent to "all reals are constructible".

Following in the footsteps of P. Erd\H{o}s and A. R\'enyi we compute the Hausdorff dimension of sets of numbers whose digits with respect to their $Q$-Cantor series expansions satisfy various statistical properties. In particular, we…

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.

We construct renormalizable, asymptotically free, four dimensional gauge theories that dynamically generate a fifth dimension.

We show some results about the Hausdorff dimension of particular minimal but not uniquely ergodic interval exchange transformations. There is an appendix which shows that typical points for two different ergodic measures of an interval…