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A useful property of Euclidean space originally shown by I. Benjamini and O. Schramm turns out to characterize doubling metric spaces.
We explore the consequences for the boundedness properties of averaging and maximal averaging operators, of the following local comparabiliity condition for measures: Intersecting balls of the same radius have comparable sizes. Since in…
We show that if $X$ is a uniformly perfect complete metric space satisfying the finite doubling property, then there exists a fully supported measure with lower regularity dimension as close to the lower dimension of $X$ as we wish.…
Measurement incompatibility is one of the cornerstones of quantum theory. This phenomenon appears in many forms, of which the concept of non-joint measurability has received considerable attention in the recent years. In order to…
This paper studies functions of bounded mean oscillation (BMO) on metric spaces equipped with a doubling measure. The main result gives characterizations for mappings that preserve BMO. This extends the corresponding Euclidean results by…
Applying the Bonk-Kleiner characterization of Ahlfors 2-regular quasispheres, we show that a metric two-sphere $X$ is a quasisphere if and only if $X$ is linearly locally connected and carries a weak metric doubling measure, i.e., a measure…
We give a short and simple polynomial estimate of the norm of weighted dyadic shift on metric space with geometric doubling, which is linear in the norm of the weight. Combined with the existence of special probability space of dyadic…
It is known that any $n$-point set in the $d$-dimensional Euclidean space $\mathbb{R}^d$, for $d = O(1)$, admits: 1) a $(1+\epsilon)$-spanner with maximum degree $\tilde{O}(\epsilon^{-d+1})$ and with lightness $\tilde{O}(\epsilon^{-d})$; 2)…
We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all…
We consider tilings and packings of $\RR^d$ by integral translates of cubes $[0,2[^d$, which are $4\ZZ^d$-periodic. Such cube packings can be described by cliques of an associated graph, which allow us to classify them in dimension $d\leq…
The aim of this paper is to study ultralimits of pointed metric measure spaces (possibly unbounded and having infinite mass). We prove that ultralimits exist under mild assumptions and are consistent with the pointed measured…
We propose a "decomposition method" to prove non-asymptotic bound for the convergence of empirical measures in various dual norms. The main point is to show that if one measures convergence in duality with sufficiently regular observables,…
Recent experiments claiming formation of quantum superposition states in near macroscopic sys- tems raise the question of how the sizes of general quantum superposition states in an interacting system are to be quantified. We propose here a…
Phenomena with a constrained sample space appear frequently in practice. This is the case e.g. with strictly positive data and with compositional data, like percentages and the like. If the natural measure of difference is not the absolute…
We consider the concept of a generalised manifold in the O(d,d) setting, i.e., in double geometry. The conjecture by Hohm and Zwiebach for the form of finite generalised diffeomorphisms is shown to hold. Transition functions on overlaps are…
We give a short proof that there exists a countable state Bernoulli measure maximizing the dimension of their images under the continued fraction expansion.
We show that there exists a family of mutually singular doubling measures on Laakso space with respect to which real-valued Lipschitz functions are almost everywhere differentiable. This implies that there exists a measure zero universal…
We address the following state comparison problem: is it possible to design an experiment enabling us to unambiguously decide (based on the observed outcome statistics) on the sameness or difference of two unknown state preparations without…
In some recent papers the classical `splitting necklace theorem' is linked in an interesting way with a geometric `pattern avoidance problem'. We explore the topological constraints on the existence of a (relaxed) measurable coloring of R^d…
Quantum measurement is universal for quantum computation. This universality allows alternative schemes to the traditional three-step organisation of quantum computation: initial state preparation, unitary transformation, measurement. In…