相关论文: Obstructions to uniformity, and arithmetic pattern…

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…

Wall-crossing phenomena are ubiquitous in many problems of algebraic geometry and theoretical physics. Various ways to encode the relevant information and the need to track the changes under the variation of parameters lead to rather…

While the prime numbers have been subject to mathematical inquiry since the ancient Greeks, the accumulated effort of understanding these numbers has - as Marcus du Sautoy recently phrased it - 'not revealed the origins of what makes the…

The main motivation for this article is to explore the connections between the existence of certain combinatorial patterns (as in van der Corputs's theorem on arithmetic progressions of length $3$) with well-known tools and theorems for…

We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi's theorem, which asserts that any subset of the integers of positive density contains progressions of…

Prime numbers appeared in contexts spanning statistical mechanics, quantum mechanics and dynamical systems. However, the mechanisms governing the irregularities observed in their sequence and linking them to physical systems remained…

We explore a surprising phenomenon in which an obstruction accelerates, rather than decelerates, a moving flexible object. It has been claimed that the right kind of discrete chain falling onto a table falls \emph{faster} than a…

We study almost sure limiting behavior of extreme and intermediate order statistics arising from strictly stationary sequences. First, we provide sufficient dependence conditions under which these order statistics converges almost surely to…

We establish mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related multiple recurrence results. A consequence of our main result is that every set of integers with positive…

This paper investigates how global decision problems over arithmetically represented domains acquire reflective structure through class-quantification. Arithmetization forces diagonal fixed points whose verification requires reflection…

We introduce the concepts of Baire Ergodicity and Ergodic Formalism, employing them to study topological and statistical attractors. Specifically, we establish the existence and finiteness of such attractors and provide applications for…

We investigate the limiting behavior of multiple ergodic averages along sparse sequences evaluated at prime numbers. Our sequences arise from smooth and well-behaved functions that have polynomial growth. Central to this topic is a…

Statistical distribution of the primes in an arithmetic progression is considered. The estimation of prime numbers is given and combinatorial methods are used to calculate the twin primes on the available interval. The distribution and…

We provide an ergodic theory framework to study statistical properties of smooth sequences over the odd alphabet {1, 3}. The arithmetic nature of this alphabet yields a partition of the subshift of smooth sequences based on their local…

Let $(X,B_X,\mu,T)$ be a measure-preserving probability system with $T$ is invertible. Suppose that $A\in B_X$ with $\mu(A)>0$ and $\epsilon>0$. For any $m\geq 1$, there exist infinitely many primes $p_0,p_1,\ldots,p_m$ with…

We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.

By means of a novel variational approach we study ergodic properties of a model of a multi lane traffic flow, considered as a (deterministic) wandering of interacting particles on an infinite lattice. For a class of initial configurations…

Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: The primes in an short interval contains many arithmetic progressions of any…

We study ergodic properties of a family of traffic maps acting in the space of bi-infinite sequences of real numbers. The corresponding dynamics mimics the motion of vehicles in a simple traffic flow, which explains the name. Using…

We give a short combinatorial proof of the classical pointwise ergodic theorem for probability measure preserving $\mathbb{Z}$-actions. Our approach reduces the theorem to a tiling problem: tightly tile each orbit by intervals with desired…