Abstract In this paper, we build some ergodic theorems involving the function $\Omega $ , where $\Omega (n)$ denotes the number of prime factors of a natural number n counted with multiplicities. As a combinatorial application, it is shown that for any $k\in \mathbb {N}$ and every $A\subset \mathbb {N}$ with positive upper Banach density, there are $a,d\in \mathbb {N}$ such that $a,a+d,\ldots, a+kd,a+\Omega(d)\in A.$

Abstract In this article, we consider a closed rank-one Riemannian manifold M without focal points. Let $P(t)$ be the set of free-homotopy classes containing a closed geodesic on M with length at most t , and $\# P(t)$ its cardinality. We obtain the following Margulis-type asymptotic estimates: $$ \begin{align*}\lim_{t\to \infty}\#P(t)/\frac{e^{ht}}{ht}=1\end{align*} $$ where h is the topological entropy of the geodesic flow. We also show that the unique measure of maximal entropy of the geodesic flow has the Bernoulli property.

Abstract The marked length spectrum (MLS) of a closed negatively curved manifold $(M, g)$ is known to determine the metric g under various circumstances. We show that, in these cases, (approximate) values of the MLS on a sufficiently large finite set approximately determine the metric. Our approach is to recover the hypotheses of our main theorems in Butt [Quantative marked length spectrum rigidity. Preprint , 2022], namely, multiplicative closeness of the MLS functions on the entire set of closed geodesics of M . We use mainly dynamical tools and arguments, but take great care to show that the constants involved depend only on concrete geometric information about the given Riemannian metrics, such as the dimension, diameter and sectional curvature bounds.

Abstract We establish a polynomial ergodic theorem for actions of the affine group of a countable field K. As an application, we deduce—via a variant of Furstenberg’s correspondence principle—that for fields of characteristic zero, any ‘large’ set $E\subset K$ contains ‘many’ patterns of the form $\{p(u)+v,uv\}$ , for every non-constant polynomial $p(x)\in K[x]$ . Our methods are flexible enough that they allow us to recover analogous density results in the setting of finite fields and, with the aid of a finitistic variant of Bergelson’s ‘colouring trick’, show that for $r\in \mathbb N$ fixed, any r-colouring of a large enough finite field will contain monochromatic patterns of the form $\{u,p(u)+v,uv\}$ . In a different direction, we obtain a double ergodic theorem for actions of the affine group of a countable field. An adaptation of the argument for affine actions of finite fields leads to a generalization of a theorem of Shkredov. Finally, to highlight the utility of the aforementioned finitistic ‘colouring trick’, we provide a conditional, elementary generalization of Green and Sanders’ $\{u,v,u+v,uv\}$ theorem.

Abstract We establish large deviation estimates related to the Darling–Kac theorem and generalized arcsine laws for occupation and waiting times of ergodic transformations preserving an infinite measure, such as non-uniformly expanding interval maps with indifferent fixed points. For the proof, we imitate the study of generalized arcsine laws for occupation times of one-dimensional diffusion processes and adopt a method of double Laplace transform.

Abstract We define a family of discontinuous maps on the circle, called Bowen–Series-like maps, for geometric presentations of surface groups. The family has $2N$ parameters, where $2N$ is the number of generators of the presentation. We prove that all maps in the family have the same topological entropy, which coincides with the volume entropy of the group presentation. This approach allows a simple algorithmic computation of the volume entropy from the presentation only, using the Milnor–Thurston theory for one-dimensional maps.