相关论文: Dimension estimates for vectorial measures with re…

This paper considers the problem of restricting the short-time Fourier transform to domains of nonzero measure in the plane and studies sampling bounds of such systems. In particular, we give a quantitative estimate for the lower sampling…

In this survey we collect and discuss some recent results on the so called "Furstenberg set problem", which in its classical form concerns the estimates of the Hausdorff dimension of planar sets containing, for any direction, a subset of an…

We address the problem of modulating a parameter onto a power-limited signal, transmitted over a discrete-time Gaussian channel and estimating this parameter at the receiver. Continuing an earlier work, where the optimal trade-off between…

We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than $1$, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In…

We establish formulas for bounds on the Haudorff measure of the intersection of certain Cantor sets with their translates. As a consequence we obtain a formula for the Hausdorff dimensions of these intersections.

A local two-dimensional resolution of singularities theorem and arguments based on the Van der Corput lemma are used to give new estimates for the decay rate of the Fourier transform of a locally defined smooth hypersurface measure in R^3,…

Suppose $0 < \alpha \leq n$, $H: \Bbb R^n \to [0,1]$ is a Lebesgue measurable function, and $A_\alpha(H)$ is the infimum of all numbers $C$ for which the inequality $\int_B H(x) dx \leq C R^\alpha$ holds for all balls $B \subset \Bbb R^n$…

In this paper we present a differential approach to photo-polarimetric shape estimation. We propose several alternative differential constraints based on polarisation and photometric shading information and show how to express them in a…

Falconer proved that there are sets $E\subset \mathbb{R}^n$ of Hausdorff dimension $n/2$ whose distance sets $\{|x-y| : x,y\in E\}$ are null with respect to Lebesgue measure. This led to the conjecture that distance sets have positive…

In this article, the Hausdorff dimension is estimated using the box-counting dimension and the information dimension. It is shown that the former is an upper bound for the Hausdorff dimension, while the latter is a lower bound for the…

The authors have recently obtained a lower bound of the Hausdorff dimension of the sets of vectors $(x_1, \ldots, x_d)\in [0,1)^d$ with large Weyl sums, namely of vectors for which $$ \left| \sum_{n=1}^{N}\exp(2\pi i (x_1 n+\ldots +x_d…

We establish variational estimates related to the problem of restricting the Fourier transform of a three-dimensional function to the two-dimensional Euclidean sphere. At the same time, we give a short survey of the recent field of maximal…

We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all first-order systems and all second-order scalar operators. In…

We study various measure theories using the classical approach and then compute the Hausdorff dimension of some simple objects and self-similar fractals. We then develop a nonstandard approach to these measure theories and examine the…

We describe the shrinking target problem for random iterated function systems which semi-conjugate to a random subshifts of finite type. We get the Hausdorff dimension of the set based on shrinking target problems with given targets. The…

This paper is devoted to dimensional reductions via the norm resolvent convergence. We derive explicit bounds on the resolvent difference as well as spectral asymptotics. The efficiency of our abstract tool is demonstrated by its…

Certain subsets of limit sets of geometrically finite Fuchsian groups with parabolic elements are considered. It is known that Jarn\'{\i}k limit sets determine a "weak multifractal spectrum" of the Patterson measure in this situation. This…

We discuss a method to estimate the measure of a compact set which is approximated using the Hausdorff distance by a sequence of compact sets. We do this by considering corresponding fattenings of the sequence of compact sets and showing…

We study Fourier frames of exponentials on fractal measures associated with a class of affine iterated function systems. We prove that, under a mild technical condition, the Beurling dimension of a Fourier frame coincides with the Hausdorff…

In the context of a metric measure Dirichlet space satisfying volume doubling and Poincar\'e inequality, we prove the parabolic Harnack inequality for weak solutions of the heat equation associated with local nonsymmetric bilinear forms. In…