Related papers: On rectifiable measures in Carnot groups: Marstran…

By a compact packing of the plane by discs, $P$, we mean a collection of closed discs in the plane with pairwise disjoint interior so that, for every disc $C\in P$, there exists a sequence of discs $D_{0},\ldots,D_{m-1}\in P$ so that each…

Let $K\subset\mathbb R^d$ be a compact subset equipped with a $\delta$-Ahlfors regular measure $\mu$. For any $\tau>1/d$ and any ``inhomogeneous'' vector $\boldsymbol{\theta}\in\mathbb R^d$, let $W_d(\psi_\tau,\boldsymbol{\theta})$ denote…

The H-type deviation, $\delta({\mathbb G})$, of a step two Carnot group ${\mathbb G}$ quantifies the extent to which ${\mathbb G}$ deviates from the geometrically and algebraically tractable class of Heisenberg-type (H-type) groups. In an…

The study of Sobolev and Poincar\'e inequalities for differential forms in Carnot groups and in the more general sub-Riemannian setting is still an open problem in its full generality. One may conjecture that, for general Carnot groups,…

We prove the following variant of Marstrand's theorem about projections of cartesian products of sets: Let $K_1,...,K_n$ Borel subsets of $\mathbb R^{m_1},... ,\mathbb R^{m_n}$ respectively, and $\pi:\mathbb R^{m_1}\times...\times\mathbb…

Let $M$ be a closed manifold and $L$ an exact magnetic Lagrangian. In this paper we proved that there exists a residual $\mathcal{G}$ of $H^{1}\left( M;\mathbb{R}\right)$ such that the property: \begin{equation*}…

Let $F$ be a non-Archimedean locally compact field and $G$ a connected reductive group defined over $F$. To any unipotent element $u$ in $G(F)$, we have associated in [L] an $F$-stratum $\boldsymbol{\mathfrak{Y}}_{F,u}$ which is a (possibly…

We study the measurable dynamics of transformations on profinite groups, in particular of those which factor through sufficiently many of the projection maps; these maps generalize the 1-Lipschitz maps on $\mathbb Z_p$.

The paper describes two possible ways of extending the definition of Haar measure to non-Hausdorff locally compact groups. The first one forces compact sets to be measurable: with this construction, a counterexample to the existence of the…

The classical Painlev\'e theorem tells that sets of zero length are removable for bounded analytic functions, while (some) sets of positive length are not. For general $K$-quasiregular mappings in planar domains the corresponding critical…

Let K be a number field. We prove that its ray class group modulo p 2 (resp. 8) if p > 2 (resp. p = 2) characterizes its p-rationality. Then we give two short, very fast PARI Programs (\S \S 3.1, 3.2) testing if K (defined by an irreducible…

A metric thickening of a given metric space $X$ is any metric space admitting an isometric embedding of $X$. Thickenings have found use in applications of topology to data analysis, where one may approximate the shape of a dataset via the…

Let G be a locally compact Hausdorff group in which every element is of finite order, and let P(G) denote the class of all regular probability measures on G. In this note, it is observed that a characterization of algebraically regular…

We prove that for a residual (and hence dense) subset $\mathcal{G}$ of Riemannian metrics on $S^{n+1}$ in the $C^{3}$ topology, no area-minimizing integral $n$-current that is a boundary admits a singular tangent cone which is linearly…

We study Riemannian manifolds $(M^n,g)$ with mean-convex boundary whose Ricci curvature is nonnegative in a spectral sense. Our first main result is a sharp spectral extension of a rigidity theorem by Kasue: we prove that under the…

Let $1 \leq m < s \leq n$ and let $A \subseteq \mathbb{R}^n$ be a Borel set of with $s$-dimensional Hausdorff measure $\mathcal{H}^s(A) > 0$. The classical Marstrand slicing theorem states that, for almost every $m$-dimensional subspace $V…

We study the Hirsch-Plotkin radical of stability groups of (general) subspace series of infinite dimensional vector spaces. We show that in countable dimension and some other cases, the HP-radical of the stability group coincides with the…

Let $f$ be a Lipschitz map from a subset $A$ of a stratified group to a Banach homogeneous group. We show that directional derivatives of $f$ act as homogeneous homomorphisms at density points of $A$ outside a $\sigma$-porous set. At…

We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small - of co-dimension at least one in…

The Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials is studied for parameters satisfying a truncation condition such that the orthogonality measure becomes discrete with support on a finite grid. For this…