Related papers: Some ergodic theorems over $k$-full numbers

We prove that there are infinitely many $n$ such that $\omega(n+k) \ll \log k$ for all integers $k \ge 2$. This improves on a result of Tao-Ter\"{a}v\"{a}inen (2025), who has $O(k)$ in place of $O(\log k)$. As corollaries, we make progress…

Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with decreasing probability of order $n^{-\alpha}$, $0 < \alpha < 1/2$. We prove that, almost surely, for every measure-preserving system…

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 $(X, {\mathcal A},\nu,\tau)$ and any $f\in…

We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on $\sigma$-finite measure spaces. We also establish corresponding…

Let $K/\mathbf Q$ be a degree $d$ extension. Inside the ring of integers $\mathcal O_K$ we define the set of $k$-free integers $\mathcal F_k$ and a natural $\mathcal O_K$-action on the space of binary $\mathcal O_K$-indexed sequences,…

In this paper, we study the ergodic theorem for infinite-dimensional quantum Markov semigroups, originally introduced by Frigerio and Verri in 1982, and its latest version developed by Carbone and Girotti in 2021. We provide a sufficient…

We present news proofs of the additivity, resolution and cofinality theorems for the algebraic $K$-theory of exact categories. These proofs are entirely algebraic, based on Grayson's presentation of higher algebraic $K$-groups via binary…

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…

We investigate uniform ergodic type theorems for additive and subadditive functions on a subshift over a finite alphabet. We show that every strictly ergodic subshift admits a uniform ergodic theorem for Banach-space-valued additive…

The celebrated Erd\H{o}s--Kac theorem says, roughly speaking, that the values of additive functions satisfying certain mild hypotheses are normally distributed. In the intervening years, similar normal distribution laws have been shown to…

We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real-rank-one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic…

We consider some natural sets of real numbers arising in ergodic theory and show that they are, respectively, complete in the classes $\mathcal D_2 (\mathbf\Pi^0_3)$ and $\mathcal D_\omega (\mathbf \Pi^0_3)$, that is, the class of sets…

In 1968, V.I. Oseledets formulated the question of convergence in the Birkhoff theorem and the multiplicative ergodic theorem for measurable cocycles over flows under the condition of integrability for each individual t. A.M. Stepin and the…

Examples are constructed of sparse subsequences of the integers for which the associated maximal averages operator is of weak type (1,1). A consequence, by transference, is that an almost everywhere L^1 -- type ergodic theorem holds for…

We analyze the elementary theory of certain fields of totally S-adic algebraic numbers that were introduced and studied by Geyer-Jarden and Haran-Jarden-Pop. In particular, we provide an axiomatization of these theories and prove their…

Recently, the dynamical and spectral properties of square-free integers, visible lattice points and various generalisations have received increased attention. One reason is the connection of one-dimensional examples such as $\mathscr…

We investigate the limiting behavior of multiple ergodic averages along sparse sequences evaluated at prime numbers. Our sequences arise from smooth and well-behaved functions that have polynomial growth. Central to this topic is a…

We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems, not…

Content of the lectures is the following. Properties of transformations equivalent to ergodicity. Birkhoff's Theorem. Properties equivalent to weak mixing. On typical properties of transformations. Lego to construct transformations. Typical…

We establish pointwise ergodic theorems for operators of Radon type along subsets of prime numbers of the form $\big\{\{ \varphi_1(n)\} < \psi(n)\big\}$. We achieve this by proving $\ell^p(\mathbb{Z})$ boundedness of $r$-variations, where…