Related papers: What is a cube?
Systems of dyadic cubes are the basic tools of harmonic analysis and geometry, and this notion had been extended to general metric spaces. In this paper, we construct systems of dyadic cubes of complete, doubling, uniformly perfect metric…
Working on doubling metric spaces, we construct generalised dyadic cubes adapting ultrametric structure. If the space is complete, then the existence of such cubes and the mass distribution principle lead into a simple proof for the…
We give a comprehensive representation of the construction of dyadic cubes in spaces of homogeneous type.
It is proposed that the physical universe is an instance of a mathematical structure which possesses a dual structure, and that this dual structure is the collection of all possible knowledge of the physical universe. In turn, the physical…
We introduce the notion of tiling spaces for metric spaces. The class of tiling spaces contains the Euclidean spaces, the middle-third Cantor set, and various self-similar spaces appearing in fractal geometry. For doubling tiling spaces, we…
Given a compact pseudo-metric space, we associate to it upper and lower dimensions, depending only on the metric. Then we construct a doubling metric for which the measure of a dillated ball is closely related to these dimensions.
In a separably connected space any two points are contained in a separable connected subset. We show a mechanism that takes a connected bounded metric space and produces a complete connected metric space whose separablewise components form…
We prove a Tb theorem on quasimetric spaces equipped with what we call an upper doubling measure. This is a property that encompasses both the doubling measures and those satisfying the upper power bound \mu(B(x,r)) \le Cr^d. Our spaces are…
A number of recent results in Euclidean Harmonic Analysis have exploited several adjacent systems of dyadic cubes, instead of just one fixed system. In this paper, we extend such constructions to general spaces of homogeneous type, making…
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…
A toric cube is a subset of the standard cube defined by binomial inequalities. These basic semialgebraic sets are precisely the images of standard cubes under monomial maps. We study toric cubes from the perspective of topological…
A metric space is indivisible if for any partition of it into finitely many pieces one piece contains an isometric copy of the whole space. Continuing our investigation of indivisible metric spaces, we show that a countable ultrametric…
We extend the notion of a dyadic grid of cubes in Euclidean space to include infinite dyadic cubes. These `tops' of a dyadic grid form a tiling of Euclidean space which is subject to the constraints similar to those arising in tiling…
We consider a topological space with its subbase which induces a coding for each point. Every second-countable Hausdorff space has a subbase that is the union of countably many pairs of disjoint open subsets. A dyadic subbase is such a…
It is shown that there exists a dihedral acute triangulation of the three-dimensional cube. The method of constructing the acute triangulation is described, and symmetries of the triangulation are discussed.
We study metric spaces that admit a conical bicombing and thus obey a weak form of non-positive curvature. Prime examples of such spaces are injective metric spaces. In this article we give a complete characterization of complete metric…
In this note, we provide equivalent definitions for fractal geometric dimensions through dyadic cube constructions. Given a metric space $X$ with finite Assouad dimension, i.e., satisfying the doubling property, we show that the…
A number of topics involving metrics and measures are discussed, including some of the special structure associated with ultrametrics.
We give a general procedure for constructing metric spaces from systems of partitions. This generalises and provides analogues of Sageev's construction of dual CAT(0) cube complexes for the settings of hyperbolic and injective metric…
We give an explicit description of cubic rings over a discrete valuation ring, as well as a description of all ideals of such rings.