Related papers: The Furstenberg set and its random version

Fix an irrational number $\alpha$. Let $X_1,X_2,\cdots$ be independent, identically distributed, integer-valued random variables with characteristic function $\varphi$, and let $S_n=\sum_{i=1}^n X_i$ be the partial sums. Consider the random…

For each odd prime $p$, we conjecture the distribution of the $p$-torsion subgroup of $K_{2n}(\mathcal{O}_F)$ as $F$ ranges over real quadratic fields, or over imaginary quadratic fields. We then prove that the average size of the…

We study eigenvalue spacings and local eigenvalue statistics for 1D lattice Schrodinger operators with Holder regular potential, obtaining a version of Minami's inequality and Poisson statistics for the local eigenvalue spacings. The main…

Real-world complex systems often comprise many distinct types of elements as well as many more types of networked interactions between elements. When the relative abundances of types can be measured well, we often observe heavy-tailed…

In this paper, we consider the Strichartz estimates for orthonormal systems in the context of randomization. Frank, Lewin, Lieb, and Seiringer first proved the orthonormal Strichartz estimates. After that, many authors have studied this…

Employing the currently discussed notion of pseudo-Hermiticity, we define a pseudo-unitary group. Further, we develop a random matrix theory which is invariant under such a group and call this ensemble of pseudo-Hermitian random matrices as…

Let X be a symmetric space of noncompact type and \Gamma a lattice in the isometry group of X. We study the distribution of orbits of \Gamma acting on the symmetric space X and its geometric boundary X(\infty). More precisely, for any y in…

Using a nonperturbative approach we examine the large frequency asymptotics of the two-point level density correlator in weakly disordered metallic grains. This allows us to study the behavior of the two-level structure factor close to the…

The Furstenberg-S\'ark\"ozy theorem asserts that the difference set $E-E$ of a subset $E \subset \mathbb{N}$ with positive upper density intersects the image set of any polynomial $P \in \mathbb{Z}[n]$ for which $P(0)=0$. Furstenberg's…

Statistical mechanics describes interaction between particles of a physical system. Particle properties of the system can be modelled with a random field on a lattice and studied at different distance scales using renormalization group…

We investigate the properties of S=1/2 Heisenberg clusters with random frustration using exact diagonalizations. This is a model for a quantum spin glass. We show that the average ground state spin is $S \propto \sqrt{N}$, where N is the…

In this article we derive quantitative uniqueness and approximation properties for (perturbations) of Riesz transforms. Seeking to provide robust arguments, we adopt a PDE point of view and realize our operators as harmonic extensions,…

We introduce a distributionally robust maximum likelihood estimation model with a Wasserstein ambiguity set to infer the inverse covariance matrix of a $p$-dimensional Gaussian random vector from $n$ independent samples. The proposed model…

A random spherical polytope $P_n$ in a spherically convex set $K \subset S^d$ as considered here is the spherical convex hull of $n$ independent, uniformly distributed random points in $K$. The behaviour of $P_n$ for a spherically convex…

Data represented by probability measures arise as empirical distributions, posterior distributions, and feature-based representations of complex objects. We study heterogeneity in a population of probability measures through the expected…

We study the long-term behavior of the iteration of a random map consisting of Lipschitz transformations on a compact metric space, independently and randomly selected according to a fixed probability measure. Such a random map is said to…

Let $\{X_n= e^{2\pi i \theta_n}\}$ be a sequence of Steinhaus random variables, where $\theta_n$ are independent and uniformly distributed on $[0,1]$. We compute the almost sure Hausdorff dimension of the images and graphs of the random…

This text is addressed to students. It is a short story about some problems in ergodic theory, both related and independent. We discuss the factorization of transformations into the product of three involutions; Furstenberg's theorem on…

We consider discrete Schr\"odinger operators on $\ell^2(\mathbb{Z})$ with bounded random but not necessarily identically distributed values of the potential. We prove spectral localization (with exponentially decaying eigenfunctions) as…

In this manuscript, we investigate the properties of systems formed by translations of an operator in the Schatten $p$-classes $\mathcal{T}^p$. We establish the existence of Schauder frames of integer translates in $\mathcal{T}^p$ for…