Related papers: Transference principles and locally symmetric spac…

The aim of the present article is to extend the Stone--Weierstrass theorem to functions ranging in a lattice normed space and order rather than topological approximation. We proceed with the machinery of Boolean valued transfer from lattice…

Computing the similarity of two point sets is a ubiquitous task in medical imaging, geometric shape comparison, trajectory analysis, and many more settings. Arguably the most basic distance measure for this task is the Hausdorff distance,…

We study optical coefficients that characterize wave propagation through layered structures called plasmonic crystals. These consist of a finite number of stacked metallic sheets embedded in dielectric hosts with a subwavelength spacing. By…

In this paper we study the problem of approximation of the $L^2$-topological invariants by their finite dimensional analogues. We obtain generalizations of the theorem of L\"uck, dealing with towers of finitely sheeted normal coverings. We…

It has been common wisdom among mathematicians that Extended Topological Field Theory in dimensions higher than two is naturally formulated in terms of n-categories with n> 1. Recently the physical meaning of these higher categorical…

It is shown that each continuous transformation $h$ from Euclidean $m$-space ($m>1$) into Euclidean $n$-space that preserves the equality of distances (that is, fulfils the implication $|x-y|=|z-w|\Rightarrow|h(x)-h(y)|=|h(z)-h(w)|$) is a…

Among plenty of applications, low-dimensional homogeneous spaces appear in cosmological models as both, classical factor spaces of multidimensional geometry and minisuperspaces in canonical quantization. Here a new tool to restrict their…

In this paper we study a family of limsup sets that are defined using iterated function systems. Our main result is an analogue of Khintchine's theorem for these sets. We then apply this result to the topic of intrinsic Diophantine…

This paper focuses on the metric properties of L\"uroth well approximable numbers, studying analogous of classical results, namely the Khintchine Theorem, the Jarn\'ik--Besicovitch Theorem, and the result of Dodson. A supplementary proof is…

We extend the Duffin--Schaeffer conjecture to the setting of systems of $m$ linear forms in $n$ variables. That is, we establish a criterion to determine whether, for a given rate of approximation, almost all or almost no $n$-by-$m$ systems…

We develop the theory of Diophantine approximation for systems of simultaneously small linear forms, which coefficients are drawn from any given analytic non-degenerate manifolds. This setup originates from a problem of Sprind\v{z}uk from…

This brief survey deals with multi-dimensional Diophantine approximations in sense of linear form and with simultaneous Diophantine approximations. We discuss the phenomenon of degenerate dimension of linear subspaces generated by the best…

A fundamental problem in manifold learning is to approximate a functional relationship in a data chosen randomly from a probability distribution supported on a low dimensional sub-manifold of a high dimensional ambient Euclidean space. The…

We show that for any $\epsilon<1$ and any $\mathcal{T}$ `drifting away from walls', Dirichlet's Theorem cannot be $\epsilon$-improved along $\mathcal{T}$ for Lebesgue almost every system of linear forms $Y$ (see the paper for definitions).…

We consider the conformal group of a space of dim n=p+q, with SO(p,q) metric. The quotient of this group by its homogeneous Weyl subgroup gives a principal fiber bundle with 2n-dim base manifold and Weyl fibers. The Cartan generalization to…

We give upper and lower bounds for Diophantine exponents measuring how well a point in the plane can be approximated by points in the orbit of a lattice $\Gamma<\mathrm{SL}_2(\mathbb{R})$ acting linearly on $\mathbb{R}^2$. Our method gives…

We consider the distribution of the orbits of the number 1 under the $\beta$-transformations $T_\beta$ as $\beta$ varies. Mainly, the size of the set of $\beta>1$ for which a given point can be well approximated by the orbit of 1 is…

Let $b\geq 2$ be an integer and $\hv$ a real number. Among other results, we compute the Hausdorff dimension of the set of real numbers $\xi$ with the property that, for every sufficiently large integer $N$, there exists an integer $n$ such…

Neural representations are not unique objects. Even when two systems realize the same downstream computation, their hidden coordinates may differ by reparameterization. A probe family intended to reveal structure already present in a…

We develop geometric versions of Rademacher and Calderon type differentiability theorems in two categories. A special case of our results is that for any Lipschitz or continuous $W^{1,p}$ Sobolev map $f$ from $[0,1]^n$ into a Euclidean…