相关论文: Patterns in sets of positive density in trees and …
We study a pinned variant of Bourgain's theorem, concerning the occurrence of affine copies of $k$-point patterns in $\mathbb{R}^d$. Focusing on the case $k=2$, which corresponds to pinned distances, we show that the classical conclusion…
In \cite{FKW} Katznelson and Weiss establish that all sufficiently large distances can always be attained between pairs of points from any given measurable subset of $\mathbb{R}^2$ of positive upper (Banach) density. A second proof of this…
We present a sharp extension of a result of Bourgain on finding configurations of $k+1$ points in general position in measurable subset of $\mathbb{R}^d$ of positive upper density whenever $d\geq k+1$ to all proper $k$-degenerate distance…
We use the language of proper CAT(-1) spaces to study thick, locally compact trees, the real, complex and quaternionic hyperbolic spaces and the hyperbolic plane over the octonions. These are rank 1 Euclidean buildings, respectively rank 1…
We establish analogues for trees of results relating the density of a set $E \subset \mathbb{N}$, the density of its set of popular differences, and the structure of $E$. To obtain our results, we formalise a correspondence principle of…
In this paper we prove equivalence of sets of axioms for non-discrete affine buildings, by providing different types of metric, exchange and atlas conditions. We apply our result to show that the definition of a Euclidean building depends…
We consider fixed-point equations for probability measures charging measured compact metric spaces that naturally yield continuum random trees. On the one hand, we study the existence/uniqueness of the fixed-points and the convergence of…
Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine…
The recent breakthrough of Guth, Iosevich, Ou, and Wang (2019) on the Falconer distance problem states that for a compact set $A\subset \mathbb{R}^2$, if the Hausdorff dimension of $A$ is greater than $\frac{5}{4}$, then the distance set…
Several results in the existing literature establish Euclidean density theorems of the following strong type. These results claim that every set of positive upper Banach density in the Euclidean space of an appropriate dimension contains…
We prove that sets with positive upper Banach density in sufficiently large dimensions contain congruent copies of all sufficiently large dilates of three specific higher-dimensional patterns. These patterns are: $2^n$ vertices of a fixed…
We prove that for random affine code tree fractals the affinity dimension is almost surely equal to the unique zero of the pressure function. As a consequence, we show that the Hausdorff, packing and box counting dimensions of such systems…
Random code-trees with necks were introduced recently to generalise the notion of $V$-variable and random homogeneous sets. While it is known that the Hausdorff and packing dimensions coincide irrespective of overlaps, their exact Hausdorff…
Using methods from ergodic theory along with properties of the Furstenberg measure we obtain conditions under which certain classes of plane self-affine sets have Hausdorff or box-counting dimensions equal to their affinity dimension. We…
We consider fixed-point equations for probability distributions on isometry classes of measured metric spaces. The construction is required to be recursive and tree-like, but we allow loops for the geodesics between points in the support of…
We study the conditions under which the isometry of spaces with metrics generated by weights given on the edges of finite trees is equivalent to the isomorphism of these trees. Similar questions are studied for ultrametric spaces generated…
It is a longstanding conjecture that given a subset $E$ of a metric space, if $E$ has finite Hausdorff measure in dimension $\alpha\ge 0$ and $\mathscr{H}^\alpha\llcorner E$ has unit density almost everywhere, then $E$ is an…
We study affine maps between affine manifolds. Even when the fibers are compact and diffeomorphic, two of them can inherit different affine structures from the source space. This leads to a fixed linear holonomy deformation theory of the…
In this paper we consider affine iterated function systems in locally compact non-Archimedean field $\mathbb{F}$. We establish the theory of singular value composition in $\mathbb{F}$ and compute box and Hausdorff dimension of self-affine…
We show that in doubling, geodesic metric measure spaces (including, for example, Euclidean space), sets of positive measure have a certain large-scale metric density property. As an application, we prove that a set of positive measure in…