Related papers: A nonlinear version of Bourgain's projection theor…

For $S_g(x,y)=x-g(y), x,y\in\mathbb{R}^n, g\in O(n),$ we investigate the Lebesgue measure and Hausdorff dimension of $S_g(A)$ given the dimension of $A$, both for general Borel subsets of $\mathbb{R}^{2n}$ and for product sets.

Let $M$ be a $d$-dimensional complete Riemannian manifold and let $\pi: SM \to M$ denote the canonical projection from the unit tangent bundle. We prove that if $E \subset SM$ is a set that invariant under the geodesic flow with Hausdorff…

We establish Marstrand-type as well as Besicovich-Federer-type projection theorems for closest-point projections onto hyperplanes in the normed space $\mathbb{R}^{n}$. In particular, we prove that if a norm on $\mathbb{R}^{n}$ is…

Let $E\subset\rr$ be a closed set of Hausdorff dimension $\alpha$. We prove that if $\alpha$ is sufficiently close to 1, and if $E$ supports a probabilistic measure obeying appropriate dimensionality and Fourier decay conditions, then $E$…

We observe that Hall's free projective extension $P \mapsto F(P)$ of partial planes is a Borel map, and use a modification of the construction introduced in [9] to conclude that the class of countable non-Desarguesian projective planes is…

We prove a restricted projection theorem for an n-2 dimensional family of projections from $\mathbb R^n$ to $\mathbb R$. The family we consider arises naturally in the context of the adjoint representation of the maximal unipotent subgroup…

We continue the work of [1, 2, 3] by analyzing the equivalence relation of bi-embeddability on various classes of countable planes, most notably the class of countable non-Desarguesian projective planes. We use constructions of the second…

We extend a well-known theorem by Jones and Makarov [JM] on the singularity of boundary distortion of planar conformal mappings. We use a different technique to recover the previous result and, moreover, generalize the result for…

This paper establishes the equivalence of the Aubin property and the strong regularity for generalized equations over $C^2$-cone reducible sets. This result resolves a long-standing question in variational analysis and extends the…

We prove an analogue for homogeneous trees and certain affine buildings of a result of Bourgain on pinned distances in sets of positive density in Euclidean spaces. Furthermore, we construct an example of a non-homogeneous tree with…

The Falconer conjecture asserts that if E is a planar set with Hausdorff dimension strictly greater than 1, then its Euclidean distance set has positive one-dimensional Lebesgue measure. We discuss the analogous question with the Euclidean…

Many results in harmonic analysis and geometric measure theory ensure the existence of geometric configurations under the largeness of sets, which are sometimes specified via the ball condition and Fourier decay. Recently,…

We extend the theory of generalized divisors so as to work on any scheme $X$ satisfying the condition $S_2$ of Serre. We define a generalized notion of Gorenstein biliaison for schemes in projective space. With this we give a new proof in a…

We prove a version of Poincar\'e's polyhedron theorem whose requirements are as local as possible. New techniques such as the use of discrete groupoids of isometries are introduced. The theorem may have a wide range of applications and can…

It is shown that the canonical formulation of the abelian BF theory in D = 3 allows to obtain topological invariants associated to curves and points in the plane. The method consists on finding the Hamiltonian on-shell of the theory coupled…

Assume that $\mathfrak A$ is a real Banach space of finite dimension $n\geq2$. Consider any Borel probability measure $\nu$ supported on the unit ball $K$ of $\mathfrak A$. We show that \[\Delta(\nu)=\int_{x \in K}\int_{ y\in…

We present a spherical version of the theorem of Blaschke that every body of constant width $w < \frac{\pi}{2}$ can be approximated as well as we wish in the sense of the Hausdorff distance by a body of constant width $w$ whose boundary…

In this paper, we study the metrical theory of Cartesian products of exact approximation sets in $\beta$-expansions. More precisely, for an integer $d \ge 2$ and real numbers $\beta_i > 1$ $(1 \le i \le d)$, we consider the set of points…

The Rolling Ball Theorem asserts that given a convex body K in Euclidean space and having a smooth surface bd(K) with all principal curvatures not exceeding c>0 at all boundary points, K necessarily has the property that to each boundary…

Let $\cS_n(\psi_1,...,\psi_n)$ denote the set of simultaneously $(\psi_1,...,\psi_n)$--approximable points in $\R^n$ and $\cSM_n(\psi)$ denote the set of multiplicatively $\psi$--approximable points in $\R^n$. Let $\cM$ be a manifold in…