Related papers: A double return times theorem

In this paper, we investigate ergodic and fractal properties of the sets $$\Lambda_y:=\Big\{n\in\mathbb{N}:\ \{u_ny\}\in I_n\Big\},$$ where $\{\cdot\}$ denotes the fractional part function, $(u_n)_{n\in\mathbb{N}}$ is an increasing sequence…

The dominated convergence theorem implies that if (f_n) is a sequence of functions on a probability space taking values in the interval [0,1], and (f_n) converges pointwise a.e., then the sequence of integrals converges to the integral of…

We introduce sufficient conditions on discrete singular integral operators for their maximal truncations to satisfy a sparse bound. The latter imply a range of quantitative weighted inequalities, which are new. As an application, we prove…

We study the pointwise convergence of some weighted averages linked to averages along cubes. We show that if $(X,\mathcal{B},\mu, T_i)$ are not necessarily commuting measure preserving systems on the same finite measure space and if $f_i,$…

The principal results proved in this expository thesis are the IP polynomial Szemer\'edi theorem for nilpotent groups and the multiple term return times theorem with nilsequence weights. It also contains extensions of the convergence…

We show that for odd-valued piecewise-constant skew products over a certain two parameter family of interval exchanges, the skew product is ergodic for a full-measure choice of parameters.

We give computable bounds on the rate of convergence of the transition probabilities to the stationary distribution for a certain class of geometrically ergodic Markov chains. Our results are different from earlier estimates of Meyn and…

Let T be a bounded linear operator acting on a complex Banach space X and (\lambda_n) a sequence of complex numbers. Our main result is that if |\lambda_n|/|\lambda_{n+1}| \to 1 and the sequence (\lambda_n T^n) is frequently universal then…

We prove strengthenings of the Birkhoff Ergodic Theorem for weakly mixing and strongly mixing measure preserving systems. We show that our pointwise theorem for weakly mixing systems is strictly stronger than the Wiener-Wintner Theorem. We…

We use techniques of proof mining to obtain a computable and uniform rate of metastability (in the sense of Tao) for the mean ergodic theorem for a finite number of commuting linear contractive operators on a uniformly convex Banach space.

Let $(S_n)_n$ be a $R^d$-valued random walk ($d\geq2$). Using Babillot's method [2], we give general conditions on the characteristic function of $S_n$ under which $(S_n)_n$ satisfies the same renewal theorem as the classical one obtained…

We study the rate of growth of ergodic sums along a sequence (a_n) of times: S_N f(x)=f(T^{a_1}x) + ... + f(T^{a_N}x). We characterize the maximal rate of growth of these ergodic sums and identify a number of sequences such as (2^n) that…

In this note we prove the a pointwise ergodic theorem for functions taking values in a separable complete CAT(0)-space, analogous to Lindenstrauss' pointwise ergodic theorem for real-valued integrable functions on a probability space…

Let $f_n(x_1, x_2, \ldots, x_n)$ denote the algebraic normal form (polynomial form) of a rotation symmetric Boolean function of degree $d$ in $n \geq d$ variables and let $wt(f_n)$ denote the Hamming weight of this function. Let $(1, a_2,…

We develop operator renewal theory for flows and apply this to infinite ergodic theory. In particular we obtain results on mixing for a large class of infinite measure semiflows. Examples of systems covered by our results include…

We establish functional limit theorems for ergodic sums of observables with power singularities for expanding circle maps. In the regime where the observables have infinite variance, we show that when rescaled by $N^{1/s}(\ln N)^\alpha$,…

We offer a new proof of the Furstenberg-Katznelson multiple recurrence theorem for several commuting probability-preserving transformations T_1, T_2, >..., T_d: \bbZ\curvearrowright (X,\S,\mu), and so, via the Furstenberg correspondence…

We prove the following extension of the Wiener--Wintner Theorem in Ergodic Theor and the Carleson Theorem on pointwise convergence of Fourier series: For all measure preserving flows $ (X,\mu , T_t)$ and $ f\in L^p (X,\mu)$, there is a set…

I prove a theorem on the precise connection of the time and phase space average of the Boltzmann equilibrium showing that the behaviour of a dynamical system with a stationary measure and a dominant equilibrium state is qualitatively…

More than a century ago, G. Kowalewski stated that for each n continuous functions on a compact interval [a,b], there exists an n-point quadrature rule (with respect to Lebesgue measure on [a,b]), which is exact for given functions. Here we…