We first present a modern simple proof of the classical ergodic Birkhoff's theorem and Bourgain's homogeneous bilinear ergodic theorem. This proof used the simple fact that the shift map on integers has a simple Lebesgue spectrum. As a consequence, we establish that the homogeneous bilinear ergodic averages along polynomials and polynomials in primes converge almost everywhere, that is, for any invertible measure preserving transformation $T$, acting on a probability space $(X, \mathcal{B}, μ)$, for any $f \in L^r(X,μ)$ , $g \in L^{r'}(X,μ)$ such that $\frac{1}{r}+\frac{1}{r'}= 1$, for any non-constant polynomials $P(n),Q(n), n \in \mathbb{Z}$, taking integer values, and for almost all $x \in X$, we have,
$$\frac{1}{N}\sum_{n=1}^{N}f(T^{P(n)}x) g(T^{Q(n)}x),$$ and $$\frac{1}{π_N}\sum_{\overset{p \leq N}{p\textrm{~~prime}}}f(T^{P(p)}x) g(T^{Q(p)}x),$$ converge. Here $π_N$ is the number of prime in $[1,N]$.
In connection with some recent results by J. Bourgain and H. Abdalaoui, M. Lemanczyk, T. De La Rue (ALR) we present a short proof of Bourgain's theorem on Mobius orthogonality property for bounded rank-one constructions. The proof of the fact (due to ALR) that bounded-recurrent constructions with Markov self-similarity have to be flat-recurrent is simplified as well.(v4 in Russian)
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 a $k$-linear pointwise ergodic theorem on an ergodic measure-preserving system implies a uniform $k$-linear nilsequence Wiener-Wintner theorem on that system. The assumption is known to hold for arbitrary systems and $k=2$ (due to Bourgain) and for distal systems and arbitrary $k$ (due to Huang, Shao, and Ye).
We establish a generalization of Bourgain double recurrence theorem by proving that for any map $T$ acting on a probability space $(X,\mathcal{A},μ)$, and for any non-constant polynomials $P, Q$ mapping natural numbers to themselves, for any $f,g \in L^2(X)$, and for almost all $x \in X$, we have $$\lim_{\bar{N} \longrightarrow +\infty} \frac{1}{N} \sum_{n=1}^{N}\boldsymbolν(n) f(T^{P(n)}x)g(T^{Q(n)}x)=0$$ where $\boldsymbolν$ is the Liouville function or the Möbius¶ function.
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.
We solve the following counting problem for measure preserving transformations. For $f\in L_+^1(μ)$, is it true that $\ds \sup_n\frac{\bN_n(f)(x)}{n} <\infty,$ where $$\ds\bN_n(f)(x)= # {k: \frac{f(T^k x)}{k}>\frac 1 n}?$$ One of the consequences is the nonvalidity of J. Bourgain's Return Time Theorem for pairs of $(L^1, L^1)$ functions.
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 …
Vector-valued maximal inequalities and multi-parameter oscillation inequalities for the polynomial ergodic averages along multi-dimensional subsets of primes
A celebrated result by Bourgain and Wierdl states that ergodic averages along primes converge almost everywhere for $L^p$-functions, $p>1$, with a polynomial version by Wierdl and Nair. Using an anti-correlation result for the von Mangoldt function due to Green and Tao we observe everywhere convergence of such averages for nilsystems and continuous functions.
We prove that the divisor function $d(n)$ counting the number of divisors of the integer $n$, is a good weighting function for the pointwise ergodic theorem. For any measurable dynamical system $(X, {\mathcal A},ν,τ)$ and any $f\in L^p(ν)$, $p>1$, the limit $$ \lim_{n\to \infty}{1\over \sum_{k=1}^{n} d(k)} \sum_{k=1}^{n} d(k)f(τ^k x)$$ exists $ν$-almost everywhere. We also obtain similar results for other arithmetical functions, like $θ(n)$ function counting the number of squarefree divisors of $n$ and the generalized Euler totient function $J_s(n)$, $s>0$. We use Bourgain's method, namely the…
We consider Bourgain's ergodic theorem regarding arithmetic averages in the cases where quantitative mixing is present in the dynamical system. Focusing on the case of the horocyclic flow, those estimates allows us to bound from above the Hausdorff dimension of the exceptional set, providing evidence towards conjectures by Margulis,Shah and Sarnak regarding equidistribution of arithmetic averages in homogeneous spaces. We also prove the existence of a uniform upper bound for the Hausdorff dimension of the exceptional set which is independent from the spectral gap.
We prove that Sarnak's conjecture holds for any infinite measure symbolic rank-one map. We further extended Bourgain-Sarnak's result, which says that the Möbius function is a good weight for the ergodic theorem, to maps acting on $σ$-finite measure spaces. We also discuss and extend Bourgain's theorem by establishing that there is a class of maps for which the Möbius disjointness property holds for any continuous bounded function. Our proof allows us to obtain an extension of Bourgain's theorem on Möbius disjointness for bounded rank one maps and a simple and self-contained proof of this fact.
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…
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…
We prove Bourgain's Return Times Theorem for a range of exponents $p$ and $q$ that are outside the duality range. An oscillation result is used to prove hitherto unknown almost everywhere convergence for the signed average analog of Bourgain's averages. As an immediate corollary we obtain a Wiener-Wintner type of result for the ergodic Hilbert series.
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. We assume without loss of generality that $μ(X)=1.$ Consider the maximal function $\dis R^*:(f, g) \in L^p\times L^q \to R^*(f, g)(x) = \sup_{n\geq 1} \frac{f(T^nx)g(T^{2n}x)}{n}.$ We obtain the following maximal inequality. For each $1<p\leq \infty$ there exists a finite constant $C_p$ such that for each $λ>0,$ and nonnegative functions $f\in L^p$ and $g\in L^1$ μ\{x: R^*(f,g)(x)>λ\} \leq C_p \bigg(\frac{\|f\|_p\|g\|_1}λ\bigg)^{1/2}. We also show that for each $α>2$ the maximal function $R^*(f,g)$…
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…
In this paper, we establish a multi-parameter version of Bellow and Losert's Wiener-Wintner type ergodic theorem for dynamical systems not necessarily being commutative. More precisely, we introduce a weight class $\mathcal{D}$, which is shown to strictly include the multi-parameter bounded Besicovitch weight class, thus including the set $$Λ_d=\left\{\{λ^{k_1}_1\dotsmλ^{k_d}_d\}_{(k_1,\dots,k_d)\in \mathbb{N}^d}:\quad (λ_1,\dots,λ_d) \in \mathbb{T}^d\right\};$$ then prove a multi-parameter Bellow and Losert's Wiener-Wintner type ergodic theorem for the class $\mathcal{D}$ and for noncommutati…
Denote by $Γ$ the set of pointwise good sequences. Those are sequences of real numbers $(a_k)$ such that for any measure preserving flow $(U_t)_{t\in \mathbb R}$ on a probability space and for any $f\in L^\infty$, the averages $\frac{1}{n} \sum_{k=1}^{n} f(U_{a_k}x) $ converge almost everywhere. We prove the following two results. [1.] If $f: (0,\infty)\to\mathbb R$ is continuous and if $\big(f(ku+v)\big)_{k\geq 1}\inΓ$ for all $u, v>0$, then $f$ is a polynomial on some subinterval $J\subset (0,\infty)$ of positive length. [2.] If $f: [0,\infty)\to\mathbb R$ is real analytic and if $\big(f(ku)…
