Related papers: Ergodic theorem and algorithmic randomness
This is a new version of our previous work. In this version, we fill a gap included in the original proof of Theorem 1.1 in our previous paper entitled "An iterative method for Kirchhoff type equations and its applications".
We give a probalistic proof of the famous Meinardus' asymptotic formula for the number of weighted partitions with weakened one of the three Meinardus' conditions, and extend the resulting version of the theorem to other two classis types…
We provide a framework for studying randomly coloured point sets in a locally compact, second-countable space on which a metrisable unimodular group acts continuously and properly. We first construct and describe an appropriate dynamical…
This is a note on a local ergodic theorem for a symmetric exclusion process defined on an infinite tower of coverings, which is associated with a finitely generated residually finite amenable group.
We present a survey of ergodic theorems for actions of algebraic and arithmetic groups recently established by the authors, as well as some of their applications. Our approach is based on spectral methods employing the unitary…
The almost sure convergence of ergodic averages in Birkhoff's pointwise ergodic theorem is known to fail in the finitely additive setting. We introduce a natural reformulation of almost sure convergence suitable for finitely additive…
The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r,p,n). As a first application, we give a…
Results concerning recurrence and ergodicity are proved in an abstract Hilbert space setting based on the proof of Khintchine's recurrence theorem for sets, and on the Hilbert space characterization of ergodicity. These results are carried…
We prove time-dependent versions of Kingman's subadditive ergodic theorem, which can be used to study stochastic processes as well as propagation of solutions to PDE in time-dependent environments.
We give a constructive proof of Carpenter's Theorem due to Kadison. Unlike the original proof our approach also yields the real case of this theorem.
We prove an effective version of the Oppenheim conjecture with a polynomial error rate. The proof is based on an effective equidistribution theorem which in turn relies on recent progress towards restricted projection problem.
We derive results on the distribution of directions of saddle connections on translation surfaces using only the Birkhoff ergodic theorem applied to the geodesic flow on the moduli space of translation surfaces. Our techniques, together…
We prove maximal inequalities for concentric ball and spherical shell averages on a general Gromov hyperbolic group, in arbitrary probability preserving actions of the group. Under an additional condition, satisfied for example by all…
We consider weighted ergodic averages indexed by primes, where the weight depends on the prime, and is a "trace function" coming from algebraic geometry. We obtain extensions the classical mean-ergodic and pointwise ergodic theorems, as…
Birkhoff's theorem states that for an ergodic automorphism, the time averages converge to the space average. Given sequence $\psi(n)\to+0$, U. Krengel proved that for any ergodic automorphism there is an indicator such that the…
Shortly after Szemer\'edi's proof that a set of positive upper density contains arbitrarily long arithmetic progressions, Furstenberg gave a new proof of this theorem using ergodic theory. This gave rise to the field of ergodic Ramsey…
The celebrated Birkhoff Ergodic Theorem asserts that, for an ergodic map, orbits of almost every point equidistributes when sampled at integer times. This result was generalized by Bourgain to many natural sparse subsets of the integers. On…
Paper withdrawn; will be replaced by revised version containing application to lattice models as well. We study hierarchical properties of Sturmian words. These properties are similar to those of substitution dynamical systems. This…
We study weighted ensemble, an interacting particle method for sampling distributions of Markov chains that has been used in computational chemistry since the 1990s. Many important applications of weighted ensemble require the computation…
In this paper, we survey physically related applications of a class of weighted quasi-Monte Carlo methods from a theoretical, deterministic perspective, and establish quantitative universal rapid convergence results via various regularity…