相关论文: Quantization dimensions of compactly supported pro…

Let $h$ be a log-correlated Gaussian field on $\R^d$, let $\gamma \in (0,\sqrt{2d}),$ let $\mu_h$ be the $\gamma$-Gaussian multiplicative chaos measure, and let $D_h$ be an exponential metric associated with $h$ satisfying certain natural…

Let $Q$ denote the space of signed measures on the Borel $\sigma$-algebra of a separable complete space $X$. We endow $Q$ with the norm $\|q\|=\sup|\int\phi dq|$, where the supremum is taken over all Lipschitz with constant 1 functions…

We introduce a fractal dimension for a metric space defined in terms of the persistent homology of extremal subsets of that space. We exhibit hypotheses under which this dimension is comparable to the upper box dimension; in particular, the…

We establish a version of the fractal uncertainty principle, obtained by Bourgain and Dyatlov in 2016, in higher dimensions. The Fourier support is limited to sets $Y\subset \mathbb{R}^d$ which can be covered by finitely many products of…

It has long been conjectured that the entropy of quantum fields across boundaries scales as the boundary area. This conjecture has not been easy to test in spacetime dimensions greater than four because of divergences in the von Neumann…

Let $(g_i)_{i=1}^M$ be a family of contractive similitudes satisfying the open set condition. Let $\nu$ be a self-similar measure associated with $(g_i)_{i=1}^M$. We study the quantization problem for the in-homogeneous self-similar measure…

In this paper we study a gradient flow approach to the problem of quantization of measures in one dimension. By embedding our problem in $L^2$, we find a continuous version of it that corresponds to the limit as the number of particles…

A new upper bound on the relative entropy is derived as a function of the total variation distance for probability measures defined on a common finite alphabet. The bound improves a previously reported bound by Csisz\'ar and Talata. It is…

This paper studies best finitely supported approximations of one-dimensional probability measures with respect to the $L^r$-Kantorovich (or transport) distance, where either the locations or the weights of the approximations' atoms are…

High-dimensional entangled states are promising candidates for increasing the security and encoding capacity of quantum systems. While it is possible to witness and set bounds for the entanglement, precisely quantifying the dimensionality…

We study the pointwise dimension for a new class of projection measures on arbitrary fractal limit sets without separation conditions. We prove that the pointwise dimension exists a.e. for this class of measures associated to equilibrium…

This work analyzes correlations arising from quantum systems subject to sequential projective measurements to certify that the system in question has a quantum dimension greater than some $d$. We refine previous known methods and show that…

In this paper we study the quantization problem for probability measures on Riemannian manifolds. Under a suitable assumption on the growth at infinity of the measure we find asymptotic estimates for the quantization error, generalizing the…

We prove preservation of $L^q$ dimensions (for $1<q\le 2$) under all orthogonal projections for a class of random measures on the plane, which includes (deterministic) homogeneous self-similar measures and a well-known family of measures…

Let $P$ be a Borel probability measure on $\mathbb R$ generated by an infinite system of similarity mappings $\{S_j : j\in \mathbb N\}$ such that $P=\sum_{j=1}^\infty \frac 1{2^j} P\circ S_j^{-1}$, where for each $j\in \mathbb N$ and $x\in…

We consider convolving a Gaussian of a varying scale epsilon against a Borel measure mu on Euclidean delta-dimensional space. The Lq norm of the result is differentiable in epsilon. We calculate this derivative and show how the upper order…

We study the generalized q-dimensions of measures supported on non-autonomous conformal attractors, which are the generalizations of Moran sets and the attractors of iterated function systems. We first prove that the critical values of…

This paper continues work begun in \cite{M1}, in which we introduced a theory of Gowers uniformity norms for singular measures on $\mathbb{R}^d$. There, given a $d$-dimensional measure $\mu$, we introduced a $(k+1)d$-dimensional measure…

We show from a categorical point of view that probability measures on certain measurable or topological spaces arise canonically as the extension of probability distributions on countable sets. We do this by constructing probability monads…

If $S$ and $T$ are infinite sequences over a finite alphabet, then the lower and upper mutual dimensions $mdim(S:T)$ and $Mdim(S:T)$ are the upper and lower densities of the algorithmic information that is shared by $S$ and $T$. In this…