arXiv subjects
Publications and source records attributed to Rongzhong Xiao.
Fix $c\in (1,2)$. Let $α$ and $β$ be two non-zero real numbers. It is shown that for any measure preserving system $(X,\mathcal{X},μ,T)$ and any $f,g\in L^{\infty}(μ)$, the limit \begin{equation*} \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}f(T^{\lfloor αn^c \rfloor}x)g(T^{\lfloor βn^c \rfloor}x) \end{equation*} exists for $μ$-a.e. $x\in X$.
Let $Ω(n)$ denote the number of prime factors of $n$, counted with multiplicity. We prove that if $A\subset\mathbb{N}^3$ has positive upper Banach density, then there are $(x,y,z)\in\mathbb{N}^3$ and $d\in\mathbb{N}$ such that $$(x,y,z),(x+d,y,z),(x,y+d,z),(x,y,z+Ω(d))\in A.$$ To establish the above result, we give an $L^2$-decoupling theorem for the triple ergodic averages $$ \frac1N\sum_{n=1}^N T_1^n f_1\,T_2^n f_2\,S^{Ω(n)}g $$ associated with three commuting transformations by isotropy factors and nilpotent structures in $\mathbb{Z}^2$-actions.
We give an affirmative answer to Owings's sumset question: for any $2$-coloring of natural numbers, there is an infinite $B\subseteq\mathbb{N}$ such that $B+B$ is monochromatic. More generally, for every $m,\ell\in\mathbb{N}$ and every $2$-coloring of $\mathbb{N}$, there is an infinite $B\subseteq\mathbb{N}$ such that $$ (m+\ell)B\cup\{mx+\ell y:x,y\in B,\ x<y\} $$ is monochromatic.
In the paper, we search for monochromatic infinite additive structures involving polynomials over $\mathbb{N}$. It is proved that for any $r\in \mathbb{N}$, any two distinct natural numbers $a,b$, and any $2$-coloring of $\mathbb{N}$, there exist two sets $B,C\subset \mathbb{N}$ with $|B|=r$ and $|C|=\infty$ such that there exists some color containing $B+aC$ and $B+bC$.
In this paper we prove that for any finite coloring of N there are lambda,rho in N such that infinitely many pairs (x,y),(u,v) in N^2 satisfy the sets {lambda x, lambda y, x y, lambda(x+y)} and {u+rho, v+rho, u v+rho, u+v} being monochromatic. Using related arguments we also give two different proofs of a special case of the Milliken--Taylor theorem.
In this paper, we study the pointwise convergence of centain continuous-time polynomial ergodic averages. Our approach is based on the topological models of measurable flows. One of the main results of this paper is as follows: Let $a\in \mathbb{R}$, $Q\in \mathbb{R}[t]$ with $\text{deg}\ Q\ge 2$. Let $(X,\mathcal{X},μ, (T^{t})_{t\in \mathbb{R}})$ and $(X,\mathcal{X},μ, (S^{t})_{t\in \mathbb{R}})$ be two measurable flows. Then for any $f_1, f_2, g\in L^{\infty}(μ)$, the limit \begin{equation*} \lim\limits_{M\to\infty}\frac{1}{M}\int_{0}^{M}f_1(T^{t}x)f_2(T^{at}x)g(S^{Q(t)}x)dt \end{equation*} exists for $μ$-a.e. $x\in X$. In particular, we are able to build a pointwise ergodic theorem involving geodesic flow and horocycle flow.
In this paper, we build some ergodic theorems involving function $Ω$, where $Ω(n)$ denotes the number of prime factors of a natural number $n$ counted with multiplicities. As a combinatorial application, it is shown that for any $k\in \mathbb{N}$ and every $A\subset \mathbb{N}$ with positive upper Banach density, there are $a,d\in \mathbb{N}$ such that $$a,a+d,\ldots,a+kd,a+Ω(d)\in A.$$
Let $G$ be an infinite countable discrete amenable group. For any $G$-action on a compact metric space $X$, it is proved that for any sequence $(G_n)_{n\ge 1}$ consisting of non-empty finite subsets of $G$ with $\lim_{n\to \infty}|G_n|=\infty$, Pinsker $σ$-algebra is a characteristic factor for $(G_n)_{n\ge 1}$. As a consequence, for a class of $G$-topological dynamical systems, positive topological entropy implies mean Li-Yorke chaos along a class of sequences consisting of non-empty finite subsets of $G$.
In this paper, we reduce pointwise convergence of polynomial ergodic averages of general measure-preserving system acted by $\mathbb{Z}^{d}$ to the case of measure-preserving system acted by $\mathbb{Z}^{d}$ with zero entropy. As an application, we can build pointwise convergence of polynomial ergodic averages for $K$-system acted by $\mathbb{Z}^{d}$.
In this paper, we extend the generalized Wiener-Wintner Theorem built by Host and Kra to the multilinear case under the hypothesis of pointwise convergence of multilinear ergodic averages. In particular, we have the following result: Let $(X,\mathcal{B},μ,T)$ be a measure preserving system. Let $a$ and $b$ be two distinct non-zero integers. Then for any $f_{1},f_{2}\in L^{\infty}(μ)$, there exists a full measure subset $X(f_{1},f_{2})$ of $X$ such that for any $x\in X(f_{1},f_{2})$, and any nilsequence $\textbf{b}=\{b_n\}_{n\in \mathbb{Z}}$, $$ \lim_{N\rightarrow \infty}\frac{1}{N}\sum_{n=0}^{N-1}b_{n}f_{1}(T^{an}x)f_{2}(T^{bn}x)$$ exists.
Let $k,a\in \mathbb{N}$ and let $p_1,\cdots,p_k\in \mathbb{Q}[n]$ with zero constant term. We show that for any finite coloring of $\mathbb{Q}$, there are non-zero $x,y\in \mathbb{Q}$ such that there exists a color which contains a set of the form $$\Big\{x,\frac{x}{y^a},x+p_{1}(y),\cdots,x+p_{k}(y)\Big\}$$ and there are non-zero $v,u\in \mathbb{Q}$ such that there exists a color which contains a set of the form $$\Big\{v,v\cdot {u^a},v+p_{1}(u),\cdots,v+p_{k}(u)\Big\}.$$