Ciprian Demeter
Let ${\bf X}=(X, Σ, m, τ)$ be a dynamical system. We prove that the bilinear series $\sideset{}{'}\sum_{n=-N}^{N}\frac{f(τ^nx)g(τ^{-n}x)}{n}$ converges almost everywhere for each $f,g\in L^{\infty}(X).$ We also give a proof along the same lines of Bourgain's analog result for averages.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We establish pointwise ergodic theorems for operators of Radon type along subsets of prime numbers of the form $\big\{\{ φ_1(n)\} < ψ(n)\big\}$. We achieve this by proving $\ell^p(\mathbb{Z})$ boundedness of $r$-variations, where $p > 1$ and $r > 2$.
Weak-type (1,1) inequality for discrete maximal functions and pointwise ergodic theorems along thin arithmetic sets
本文建立了沿稀疏算术集合的离散极大函数弱型 (1,1) 界，由此获得 $L^1$ 逐点收敛结果作为反例，并推导出 $L^p$ 上的多参数逐点遍历定理。
For any dynamical system, we show that higher variation-norms for the sequence of ergodic bilinear averages of two functions satisfy a large range of bilinear Lp estimates. It follows that, with probability one, the number of fluctuations along this sequence may grow at most polynomially with respect to (the growth of) the underlying scale. These results strengthen previous works of Lacey and Bourgain where almost surely convergence of the sequence was proved (which is equivalent to the qualitative statement that the number of fluctuations is finite at each scale). Via transference, the proof …
We provide an exposition of the proofs of Bourgain's polynomial ergodic theorems. The focus is on the motivation and intuition behind his arguments.
Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with decreasing probability of order $n^{-α}$, $0 < α< 1/2$. We prove that, almost surely, for every measure-preserving system $(X,T)$ and every $f \in L^1(X)$ orthogonal to the invariant factor, the modulated, random averages \[ \sup_{b} \Big| \frac{1}{N} \sum_{n = 1}^N b(n) T^{a_{n}} f \Big| \] converge to $0$ pointwise almost everywhere, where the supremum is taken over a set of bounded functions with certain uniform approximation properties.
Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with probability $n^{-a}$, $0 < a < 1/2$, and let $p(n) = n^{1+ε}$, $0 < ε< 1$. We prove that, almost surely, for every measure-preserving system $(X,T)$ and every $f \in L^1(X)$ the modulated, random averages \[ \frac{1}{N} \sum_{n = 1}^N e(p(n)) T^{a_n(ω)} f\] converge to $0$ pointwise almost everywhere.
We prove a.e. convergence of continuous-time quadratic averages with respect to two commuting $\mathbb{R}$-actions, coming from a single jointly measurable measure-preserving $\mathbb{R}^2$-action on a probability space. The key ingredient of the proof comes from recent work on multilinear singular integrals; more specifically, from the study of a curved model for the triangular Hilbert transform.
The circle method and pointwise ergodic theorems
本文通过圆方法探讨了多项式遍历平均算子在几乎处处收敛与范数收敛之间的本质差异，并指出不同工具可能带来更深入的洞察。
Pointwise ergodic theorems for non-conventional bilinear averages along $(\lfloor n^c\rfloor,-\lfloor n^c\rfloor)$
本文证明了在概率动力系统中，沿$(\lfloor n^c\rfloor,-\lfloor n^c\rfloor)$等$c$-正则变化稀疏轨道的双线性遍历平均的逐点收敛性，这是首个具有调制不变性的确定性稀疏轨道逐点结果。
We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on $σ$-finite measure spaces. We also establish corresponding maximal inequalities on $L^p$ for $1<p\leq \infty$ and $ρ$-variational inequalities on $L^2$ for $2<ρ<\infty$. This gives an affirmative answer to the Furstenberg-Bergelson-Leibman conjecture in the linear case for all polynomial ergodic averages in discrete nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fouri…
Multi-Frequency Oscillation Estimates Arising in Pointwise Ergodic Theory
本研究证明了Bourgain所研究的极大傅里叶乘子算子变分形式的最优$L^p$估计，并将其实变量多项式整数部分遍历平均逐点收敛的结果推广到更宽函数类，给出了对应收敛的$p$值范围。
We prove that if $μ_n$ are probability measures on $Z$ such that $\hat μ_n$ converges to 0 uniformly on every compact subset of $(0,1)$, then there exists a subsequence $\{n_k\}$ such that the weighted ergodic averages corresponding to $μ_{n_k}$ satisfy a pointwise ergodic theorem in $L^1$. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along $n^2+ \lfloor ρ(n)\rfloor$ for a slowly growing function $ρ$. Under some monotonicity assumptions, the rate of growth of $ρ'(x)$ determines the existence of…
We investigate the Bilinear Hilbert Transform in the plane and the pointwise convergence of bilinear averages in Ergodic theory, arising from $\Z^2$ actions. Our techniques combine novel one and a half dimensional phase-space analysis with more standard one dimensional theory.
We establish the first pointwise ergodic theorems along thin sets of prime numbers; a set with zero density with respect to the primes. For instance we will be able to achieve this with the Piatetski-Shapiro primes. Our methods will be robust enough to solve the ternary Goldbach problem for some thin sets of primes.
Pointwise ergodic theorems along fractional powers of primes
该研究建立了沿素数分数次幂及更一般c-正则变化函数序列的非传统遍历平均的逐点收敛性，给出多参数振荡估计与对应定理，借助指数和估计得到Waring型结果，并在Riemann zeta函数无零点带假设下探讨L^1上的条件性结果。
We establish convergence in norm and pointwise almost everywhere for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages \[ A_N(f,g)(x) := \frac{1}{N} \sum_{n =1}^N f(T^nx) g(T^{P(n)}x)\] as $N \to \infty$, where $T \colon X \to X$ is a measure-preserving transformation of a $σ$-finite measure space $(X,μ)$, $P(\mathrm{n}) \in \mathbb Z[\mathrm{n}]$ is a polynomial of degree $d \geq 2$, and $f \in L^{p_1}(X), \ g \in L^{p_2}(X)$ for some $p_1,p_2 > 1$ with $\frac{1}{p_1} + \frac{1}{p_2} \leq 1$. We also establish an $r$-variational inequality for these avera…
In this survey we review useful tools that naturally arise in the study of pointwise convergence problems in analysis, ergodic theory and probability. We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point of view of oscillation estimates in both the single and several parameter settings. We establish a number of new oscillation inequalities and give new proofs for known results with elementary arguments.
In this paper, we establish a noncommutative analogue of Calderón's transference principle, which allows us to deduce noncommutative ergodic maximal inequalities from the special case---operator-valued maximal inequalities. As applications, we deduce dimension-free estimates of noncommutative Wiener's maximal ergodic inequality and noncommutative Stein-Calderón's maximal ergodic inequality over Euclidean spaces. We also show the corresponding individual ergodic theorems. To show Wiener's pointwise ergodic theorem, we construct a dense subset on which pointwise convergence holds following a som…
Pointwise Convergence of Random Ergodic Averages at the \(L^1\) Endpoint
本文证明沿二次增长随机序列的遍历平均在 L^1 端点几乎处处收敛，结合 Christ 调和分析与二次形式随机采样，克服均匀性障碍。
We generalize the respective ``double recurrence'' results of Bourgain and of the second author, which established for pairs of $L^{\infty}$ functions on a finite measure space the a.e. convergence of the discrete bilinear ergodic averages and of the discrete bilinear Hilbert averages defined by invertible measure-preserving point transformations. Our generalizations are set in the context of arbitrary sigma-finite measure spaces and take the form of a.e. convergence of such discrete averages, as well as of their continuous variable counterparts, when these averages are defined by Lebesgue spa…
Pointwise convergence of bilinear polynomial averages over the primes
研究证明了在素数上的双线性多项式平均在特定条件下几乎处处收敛，并结合了多项工具和估计方法。
The shifted bilinear Hilbert transform
本文证明了移位双线性Hilbert变换的$ L^p $估计，给出了移位大小的多项对数界，并应用该结果得到双线性ergodic平均的$r$-变分估计、尖锐的双线性Hörmander乘法定理以及双线性奇异积分的$\log$-Dini定理。
Polynomial Ergodic Averages Along Short Intervals
该研究证明了短区间上次数$d\geq2$、起始时间双间隙的多项式轨道在$1<p<\infty$且$c>(d-1)/d$时的$L^p$变分估计及几乎处处收敛，还指出端点$L^1$不成立，且密集起始时间序列具强扫出性质。
Pointwise Convergence of Ergodic Averages Along Hardy Field Sequences
本文证明了在配备交换保测变换的测度空间上，沿Hardy field序列的遍历平均在 $p>1$ 时的逐点收敛性，通过建立长变分不等式和极大不等式实现，并对特定函数类给出了完全变分估计及指数和收敛速率的定量界。
Polynomial ergodic theorems in the spirit of Dunford and Zygmund
本文证明了多参数多项式遍历平均在$ L^p $空间中的收敛性，并探讨了其与Bellow-Furstenberg问题变体的联系。
Fractionally modulated discrete Carleson's Theorem and pointwise Ergodic Theorems along certain curves
本文研究了分数调制离散卡尔松算子和沿特定曲线的点wise遍历平均的有界性与几乎处处收敛性，使用圆法变种分析指数和。
Variational Estimates for Bilinear Ergodic Averages Along Sublinear Sequences
本文证明了沿次线性序列的双线性遍历平均在任意保测系统上的长变分估计，覆盖完整预期范围并在端点处达到最佳，并由此导出双线性极大估计。
Pointwise convergence of polynomial multiple ergodic averages along the primes
研究沿素数加权的多项式多重遍历平均的点态几乎处处收敛性，通过发展多重线性圆法，结合Weyl型不等式和p进L^q改进估计，证明了收敛性并得到了尖锐的r变差估计。
Averages with the Gaussian divisor: Weighted Inequalities and the Pointwise Ergodic Theorem
本文研究高斯除数函数加权平均的逐点遍历定理，通过建立改进不等式和加权极大不等式，证明对 $L^p$（$1<p\leq\infty$）函数相应极限几乎处处收敛。
Pointwise convergence of ergodic averages along quadratic bracket polynomials
本文证明了沿 $(n\lfloor n\sqrt{k}\rfloor)$ 形式轨道的遍历平均的逐点收敛性，通过改进的圆法和 Green 与 Tao 的多项式轨道定量行为结果解决了超出小分母有理数的算术障碍。
A Unified Approach to Two Pointwise Ergodic Theorems: Double Recurrence and Return Times
本研究提出统一方法，结合停时与度量熵，将Bourgain双重递归定理与返回时间定理推广到Kronecker序列整数部分，证明两类双线性遍历平均的几乎处处收敛性且收敛集对所有有理数γ一致。
We prove pointwise variational Lp bounds for a bilinear Fourier integral operator in a large but not necessarily sharp range of exponents. This result is a joint strengthening of the corresponding bounds for the classical Carleson operator, the bilinear Hilbert transform, the variation norm Carleson operator, and the bi-Carleson operator. Terry Lyon's rough path theory allows for extension of our result to multilinear estimates. We consider our result a proof of concept for a wider array of similar estimates with possible applications to ordinary differential equations.
In this note we introduce a sequence of bilinear operators that unify ergodic averages and backward martingales in a nontrivial way. We establish its convergence in a range of $L^p$-norms and leave its a.s. convergence as an open problem. This problem shares some similarities with a well-known unresolved conjecture on a.s. convergence of double ergodic averages with respect to two commuting transformations.
Let $(X,\mathcal{B}, μ, T)$ be an ergodic dynamical system on a non-atomic finite measure space. Consider the maximal function $\dis R^*:(f, g) \in L^1\times L^1 \to R^*(f, g)(x) = \sup_{n} \frac{f(T^nx)g(T^{2n}x)}{n}.$ We show that there exist $f$ and $g$ such that $R^*(f, g)(x)$ is not finite almost everywhere. Two consequences are derived. The bilinear Hardy--Littlewood maximal function fails to be a.e. finite for all functions $(f, g)\in L^1\times L^1.$ The Furstenberg averages do not converge for all pairs of $(L^{1},L^{1})$ functions, while by a result of J. Bourgain these averages conve…
Motivated by Bourgain's work on pointwise ergodic theorems, and the work of Stein and Stein-Wainger on maximally modulated singular integrals without linear terms, we prove that the maximally monomially modulated discrete Hilbert transform, \[ \mathcal{C}_df(x) := \sup_λ\left| \sum_{m \neq 0} f(x-m) \frac{e^{2πi λm^d}}{m} \right| \] is bounded on all $\ell^p, \ 2 - \frac{1}{d^2 + 1} < p < \infty$, for any $d \geq 2$. We also establish almost everywhere pointwise convergence of the modulated ergodic Hilbert transforms (as $λ\to 0$) \[ \sum_{m \neq 0} T^m f(x) \cdot \frac{e^{2πi λm^d}}{m} \] for…
It is shown that the homogeneous ergodic bilinear averages with Möbius or Liouville weight converge almost surely to zero, that is, if $T$ is a map acting on a probability space $(X,\mathcal{A},μ)$, and $a,b \in \mathbb{Z}$, then for any $f,g \in L^2(X)$, for almost all $x \in X$, $${\frac{1}{N}}\sum_{n=1}^{N} ν(n) f(T^{an}x) g(T^{bn}x) \longrightarrow 0, \text{ as } N \longrightarrow +\infty, $$ where $ν$ is the Liouville function or the Möbius function. We further obtain that the convergence almost everywhere holds for the short interval with the help of Zhan's estimation. Also our proof yie…
Pointwise Convergence of Ergodic Averages Along Integer Cantor Sets
研究沿整数康托集的遍历平均的逐点收敛，通过证明相关遍历平均在特定条件下几乎处处收敛，解决了康托测度在自相似尺度下的稀疏微分问题。
Let $L^2(X,Σ,μ,τ)$ be a measure-preserving system, with $τ$ a $\mathbb{Z}$-action. In this note, we prove that the ergodic averages along integer-valued polynomials, $P(n)$, \[ M_N(f):= \frac{1}{N}\sum_{n \leq N} τ^{P(n)} f \] converge pointwise for $f \in L^2(X)$. We do so by proving that, for $r>2$, the $r$-variation, $\mathcal{V}^r(M_N(f))$, extends to a bounded operator on $L^2$. We also prove that our result is sharp, in that $\mathcal{V}^2(M_N(f))$ is an unbounded operator on $L^2$.
We show that $ { ω}(n)$ and $ { Ω}(n)$, the number of distinct prime factors of $n$ and the number of distinct prime factors of $n$ counted according to multiplicity are good weighting functions for the pointwise ergodic theorem in $L^{1}$. That is, if $g$ denotes one of these functions and $S_{g,K}=\sum_{n\leq K}g(n)$ then for every ergodic dynamical system $(X, { { \cal A } },μ, { τ})$ and every $f\in L^{1}(X)$ $$\lim_{K\to { \infty }} \frac{1}{S_{g,K}}\sum_{n=1}^{K} g(n)f( { τ}^{n}x)=\int_{X}fdμ\text{ for $μ$ a.e. }x\in X. $$ This answers a question raised by C. Cuny and M. Weber who showed…
A.E. Convergence vs Boundedness
本文将Stein最大定理扩展到双线性设置，证明了双线性算子的弱类型有界性和强类型有界性，以及双线性尾算子和ergodic平均的几乎处处收敛性。
On the speed of convergence in the ergodic theorem for shift operators
本文研究移位算子遍历平均的逐点收敛速度，证明在适当条件下其几乎处处收敛到零的速度本质上为 $N^{-1/2}$，并给出在环面自同态相关移位中的应用。
A Roth theorem in $\mathbb R^2$ and a related ergodic theorem
证明二维多项式模式在平面上的定量罗斯定理，得到相关多项式遍历平均的逐点收敛结果，新的双线性索伯列夫改进估计是主要分析工具。
It is well-known that estimates for maximal operators and questions of pointwise convergence are strongly connected. In recent years, convergence properties of so-called `non-conventional ergodic averages' have been studied by a number of authors, including Assani, Austin, Host, Kra, Tao, and so on. In particular, much is known regarding convergence in $L^2$ of these averages, but little is known about pointwise convergence. In this spirit, we consider the pointwise convergence of a particular ergodic average and study the corresponding maximal trilinear operator (over $\mathbb{R}$, thanks to …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。