Related papers: Counting and Joint equidistribution of approximate…

Point counting estimates are a key stepping stone to various results in metric Diophantine approximation. In this paper we use the quantitative non-divergence estimates originally developed by Kleinbock and Margulis to improve lower bounds…

We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We obtain basic results, both probabilistic and deterministic, draw connections to…

A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It…

We prove new results, related to the Littlewood and Mixed Littlewood conjectures in Diophantine approximation.

The paper is mostly a survey on recent results in Diophantine approximation, with emphasis on properties of exponents measuring various notions of Diophantine <approximation.

We study non-expanding random walks on the space of affine lattices and establish a new classification theorem for stationary measures. Further, we prove a theorem that relates the genericity with respect to these random walks to Birkhoff…

We study the convergence of the new family of mimetic finite difference schemes for linear diffusion problems recently proposed in [38]. In contrast to the conventional approach, the diffusion coefficient enters both the primary mimetic…

We propose a numerical integrator for determining low-rank approximations to solutions of large-scale matrix differential equations. The considered differential equations are semilinear and stiff. Our method consists of first splitting the…

We study some problems in metric Diophantine approximation over local fields of positive characteristic.

In this paper we prove inequalities for multiplicative analogues of Diophantine exponents, similar to the ones known in the classical case. Particularly, we show that a matrix is badly approximable if and only if its transpose is badly…

We consider a generalization of the so-called elephant random walk by introducing multiple elephants moving along the integer line, $\mathbb{Z}$. When taking a new step, each elephant considers not only its own previous steps but also the…

Following Schmidt, Thurnheer and Bugeaud-Kristensen, we study how Dirichlet's theorem on linear forms needs to be modified when one requires that the vectors of coefficients of the linear forms make a bounded acute angle with respect to a…

We study properties of Diophantine exponents of lattices and so-called related "weak" uniform approximations introduced in recent papers by Oleg German, in the simplest two-dimensional case. In contrast to the multidimensional case, in the…

We resolve problems posed by Kesten and Erd\"os-Sz\"usz-Tur\'an on probabilistic Diophantine approximation via methods of homogeneous dynamics. Our methods allows us to generalize the problem to the setting of general measure-valued…

Using purely combinatorial means we obtain results on simultaneous Diophantine approximation modulo 1 for systems of polynomials with real coefficients and no constant term.

We study the problem of Diophantine approximation on lines in R^2 with prime numerator and denominator.

We present a survey of ergodic theorems for actions of algebraic and arithmetic groups recently established by the authors, as well as some of their applications. Our approach is based on spectral methods employing the unitary…

Let $F \subseteq [0,1]$ be a set that supports a probability measure $\mu$ with the property that $ |\widehat{\mu}(t)| \ll (\log |t|)^{-A}$ for some constant $ A > 0 $. Let $\mathcal{A}= (q_n)_{n\in \mathbb{N}} $ be a sequence of natural…

In time series analysis, statistics based on collections of estimators computed from sub-samples play a crucial role in an increasing variety of important applications. Proving results about the joint asymptotic distribution of such…

In this paper we initiate a new approach to studying approximations by rational points to points on smooth submanifolds of $\mathbb{R}^n$. Our main result is a convergence Khintchine type theorem for arbitrary nondegenerate submanifolds of…