Related papers: Uniform syndeticity in multiple recurrence

Two proofs of the Koml\'os-Major-Tusn\'ady embedding theorems, one for the uniform empirical process and one for the simple symmetric random walk, are given. More precisely, what are proved are the univariate coupling results needed in the…

We prove that, given $\epsilon>0$ and $k\geq 1$, there is an integer $n$ such that the following holds. Suppose $G$ is a finite group and $A\subseteq G$ is $k$-stable. Then there is a normal subgroup $H\leq G$ of index at most $n$, and a…

For a metric space $(A,d)$, and a set $\Sigma$ of equations, some quantities are introduced that measure the size of discontinuities that must occur in operations satisfying $\Sigma$ (identically) on $A$. We are able to evaluate these…

Detecting and measuring repetitiveness of strings is a problem that has been extensively studied in data compression and text indexing. However, when the data are structured in a non-linear way, like in the context of two-dimensional…

We prove the following statement: Let $X=\text{SL}_n(\mathbb{Z})\backslash \text{SL}_n(\mathbb{R})$, and consider the standard action of the diagonal group $A<\text{SL}_n(\mathbb{R})$ on it. Let $\mu$ be an $A$-invariant probability measure…

Grothendieck's theorem asserts that every continuous linear operator from $\ell_1$ to $\ell_2$ is absolutely $(1,1)$-summing. This kind of result is commonly called coincidence result. In this paper we investigate coincidence results in the…

We give conditions for when two Euler products are the same given that they satisfy a functional equation and their coefficients are not too large and do not differ from each other by too much. Additionally, we prove a number of…

Consider a mixing dynamical systems $([0,1], T, \mu)$, for instance a piecewise expanding interval map with a Gibbs measure $\mu$. Given a non-summable sequence $(m_k)$ of non-negative numbers, one may define $r_k (x)$ such that $\mu (B(x,…

We study the behavior of the random walk on the infinite cluster of independent long range percolation in dimensions $d=1,2$, where $x$ and $y$ a re connected with probability $\sim\beta/\|x-y\|^{-s}$. We show that when $d<s<2d$ the walk is…

We prove a generalization of van der Corput's Difference Theorem in the theory of uniform distribution by establishing a connection with unitary operators that have Lebesgue spectrum. This allows us to show, for example, that if $(x_n)_{n =…

Using the local approach to the global structure of a symmetric space $E$ we establish a relationship between strict $K$- monotonicity, lower (resp. upper) local uniform $K$-monotonicity, order continuity and the Kadec-Klee property for…

Given sequence of measure preserving transformations $\{U_k:\,k=1,2,\ldots, n\}$ on a measurable space $(X,\mu)$. We prove a.e. convergence of the ergodic means \begin{equation} \frac{1}{s_1\cdots…

We introduce the notion of {\bf a}-walk $S(n)=a_1 X_1+\dots+a_n X_n$, based on a sequence of positive numbers ${\bf a}=(a_1,a_2,\dots)$ and a Rademacher sequence $X_1,X_2,\dots$. We study recurrence/transience (properly defined) of such…

We establish asymptotic formulas for sums of reciprocals of primes in arithmetic progressions, generalizing recent results on multiple Mertens evaluations by Tenenbaum, Qi, and Hu. Specifically, for any fixed constant $K>0$, we derive…

In this paper we show how polynomial walks can be used to establish a twisted recurrence for sets of positive density in $\mathbb{Z}^d$. In particular, we prove that if $\Gamma \leq \operatorname{GL}_d(\mathbb{Z})$ is finitely generated by…

We prove that any II$_1$ factor $L^{\infty}(X)\rtimes\Gamma$ arising from a compact, free, ergodic, probability measure preserving action of a countable group $\Gamma$ with positive first $\ell^2$-Betti number, has a unique group measure…

Let $\Omega$ be an irreducible bounded symmetric domain of rank $r$ in $\mathbb C^d.$ Let $\mathbb K$ be the maximal compact subgroup of the identity component $G$ of the biholomorphic automorphism group of the domain $\Omega$. The group…

The parametrization theorem is derived in a flat nD pseudo-complex affine space. The pseudo-complex hyperbolic space accomodates n-number of uncompactified time-like extra dimensions with sugnature (s,r), where s and r are the numbers of…

Let the random variable $Z_{n,k}$ denote the number of increasing subsequences of length $k$ in a random permutation from $S_n$, the symmetric group of permutations of $\{1,...,n\}$. We show that $Var(Z_{n,k_n})=o((EZ_{n,k_n})^2)$ as $…

The aim of this paper is to develop analytic techniques to deal with certain monotonicity of combinatorial sequences. (1) A criterion for the monotonicity of the function $\sqrt[x]{f(x)}$ is given, which is a continuous analog for one…