Related papers: Ergodic recurrence and bounded gaps between primes

For any measure preserving system $(X,\mathcal{X},\mu,T)$ and $A\in\mathcal{X}$ with $\mu(A)>0$, we show that there exist infinitely many primes $p$ such that $\mu\bigl(A\cap T^{-(p-1)}A\cap T^{-2(p-1)}A\bigr) > 0$ (the same holds with…

Let $(X,\mu,T_1,...,T_l)$ be a measure-preserving system with those $T_i$ are commuting. Suppose that the polynomials $p_1(t),...,p_{l}(t)\in\Z[t]$ with $p_j(0)=0$ have distinct degrees. Then for any $\epsilon>0$ and $A\subseteq X$ with…

Let $p_n$ denote the $n$th prime and $g_n:=p_{n+1}-p_n$ the $n$th prime gap. We demonstrate the existence of infinitely many values of $n$ for which $g_n>g_{n+1}>\cdots>g_{n+m}$ with $m\gg \log\log\log n$ and similarly for the reversed…

For each $m\geq 1$, there exist infinitely many primes $p_1<p_2<\ldots<p_{m+1}$ such that $p_{m+1}-p_1=O(m^4e^{8m})$ and $p_j+2$ has at most $\frac{16m}{\log 2}+\frac{5\log m}{\log 2}+37$ prime divisors for each $j$.

Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n}, $$…

Let $\mathcal{P}$ be an (unbounded) countable multiset of primes, let $G=\bigoplus_{p\in P}\mathbb{F}_p$. We study the $k$'th universal characteristic factors of an ergodic probability system $(X,\mathcal{B},\mu)$ with respect to some…

Let $(X,\mathcal{A}, \mu)$ be a probability measure space and let $T_i,$ $1\leq i\leq H,$ be invertible bi measurable measure preserving transformations on this measure space. We give a sufficient condition for the product of $H$ bounded…

Suppose that $1<c<9/8$. For any $m\geq 1$, there exist infinitely many $n$ such that $$ \{[n^c],\ [(n+1)^c],\ \ldots,\ [(n+k_0)^c]\} $$ contains at least $m+1$ primes, if $k_0$ is sufficiently large (only depending on $m$).

We introduce the notions of over- and under-independence for weakly mixing and (free) ergodic measure preserving actions and establish new results which complement and extend the theorems obtained in [BoFW] and [A]. Here is a sample of…

We provide various counter examples for quantitative multiple recurrence problems for systems with more than one transformation. We show that $\bullet$ There exists an ergodic system $(X,\mathcal{X},\mu,T_1,T_2)$ with two commuting…

Let $(X, \mathcal{B},\mu,T)$ be an ergodic measure preserving system, $A \in \mathcal{B}$ and $\epsilon>0$. We study the largeness of sets of the form \begin{equation*} \begin{split} S = \left\{ n\in\mathbb{N}\colon\mu(A\cap…

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…

We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemeredi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements…

Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with decreasing probability of order $n^{-\alpha}$, $0 < \alpha < 1/2$. We prove that, almost surely, for every measure-preserving system…

We establish the existence of infinitely many \emph{polynomial} progressions in the primes; more precisely, given any integer-valued polynomials $P_1, >..., P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0) = ... = P_k(0) = 0$ and any $\eps…

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that…

We show that if $(X, \mu, T)$ is a probability measure-preserving dynamical system, and $\mathscr{P}$ is a countable partition of $(X, \mu)$, then the limit $$ \lim_{n, k \to \infty} \mathbb{E} \left[ \frac{1}{k} \sum_{j = 0}^{k - 1} f…

Using Duke's large sieve inequality for Hecke Gr{\"o}ssencharaktere and the new sieve methods of Maynard and Tao, we prove a general result on gaps between primes in the context of multidimensional Hecke equidistribution. As an application,…

The Khintchine recurrence theorem asserts that on a measure preserving system, for every set $A$ and $\varepsilon>0$, we have $\mu(A\cap T^{-n}A)\geq \mu(A)^2-\varepsilon$ for infinitely many $n\in \mathbb{N}$. We show that there are…

Let $(X,\mathcal{B},\mu)$ be a probability space and let $T_1,..., T_l$ be $l$ commuting invertible measure preserving transformations \linebreak of $X$. We show that if $T_1^{c_1} ... T_l^{c_l}$ is ergodic for each $(c_1,...,c_l)\neq…