Abstract Approximate lattices of Euclidean spaces, also known as Meyer sets, are aperiodic subsets with fascinating properties. In general, approximate lattices are defined as approximate subgroups of locally compact groups that are discrete and have finite co-volume. A theorem of Lagarias [Meyer’s concept of quasicrystal and quasiregular sets. Comm. Math. Phys. 179 (2) (1996), 365–376] provides a criterion for discrete subsets of Euclidean spaces to be approximate lattices. It asserts that if a subset X of $\mathbb {R}^n$ is relatively dense and $X - X$ is uniformly discrete, then X is an approximate lattice. We prove two generalizations of Lagarias’ theorem: when the ambient group is amenable and when it is a higher-rank simple algebraic group over a characteristic $0$ local field. This is a natural counterpart to the recent structure results for approximate lattices in non-commutative locally compact groups. We also provide a reformulation in dynamical terms pertaining to return times of cross-sections. Our method relies on counting arguments involving the so-called periodization maps, ergodic theorems and a method of Tao regarding small doubling for finite subsets. In the case of simple algebraic groups over local fields, we moreover make use of deep superrigidity results due to Margulis and to Zimmer.

Abstract We prove that when the Aubry set for a Lipschitz continuous potential is a subshift of finite type, then the pressure function converges exponentially fast to its asymptote as the temperature goes to 0. The speed of convergence turns out to be the unique eigenvalue for the matrix whose entries are the costs between the different irreducible pieces of the Aubry set. For a special case of Walters potential, we show that perturbations of that potential that go faster to zero than the pressure do not change the selection, neither for the subaction nor for the limit measure, a zero temperature.

Abstract We solve the dry ten Martini problem for the unitary almost-Mathieu operator with Diophantine frequencies in the non-critical regime.

Abstract Given $\beta>1$ , let $T_\beta $ be the $\beta $ -transformation on the unit circle $[0,1)$ , defined by $T_\beta (x)=\beta x-\lfloor \beta x\rfloor $ . For each $t\in [0,1)$ , let $K_\beta (t)$ be the survivor set consisting of all $x\in [0,1)$ whose orbit $\{T^n_\beta (x): n\ge 0\}$ never enters the interval $[0,t)$ . Kalle et al [ Ergod. Th. & Dynam. Sys. 40 (9) (2020), 2482–2514] considered the case $\beta \in (1,2]$ . They studied the set-valued bifurcation set $\mathscr {E}_\beta :=\{t\in [0,1): K_\beta (t')\ne K_\beta (t)~\text { for all } t'>t\}$ and proved that the Hausdorff dimension function $t\mapsto \dim _H K_\beta (t)$ is a non-increasing Devil’s staircase. In a previous paper [ Ergod. Th. & Dynam. Sys. 43 (6) (2023), 1785–1828], we determined, for all $\beta \in (1,2]$ , the critical value $\tau (\beta ):=\min \{t>0: \eta _\beta (t)=0\}$ . The purpose of the present article is to extend these results to all $\beta>1$ . In addition to calculating $\tau (\beta )$ , we show that: (i) the function $\tau : \beta \mapsto \tau (\beta )$ is left-continuous on $(1,\infty )$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $\tau $ has no downward jumps; and (iii) there exists an open set $O\subset (1,\infty )$ , whose complement $(1,\infty )\setminus O$ has zero Hausdorff dimension, such that $\tau $ is real-analytic, strictly convex, and strictly decreasing on each connected component of O . We also prove several topological properties of the bifurcation set $\mathscr {E}_\beta $ . The key to extending the results from $\beta \in (1,2]$ to all $\beta>1$ is an appropriate generalization of the Farey words that are used to parameterize the connected components of the set O . Some of the original proofs from the above-mentioned papers are simplified.

Abstract Let $\gamma _{n}= O (\log ^{-c}n)$ and let $\nu $ be the infinite product measure whose n th marginal is Bernoulli $(1/2+\gamma _{n})$ . We show that $c=1/2$ is the threshold, above which $\nu $ -almost every point is simply Poisson generic in the sense of Peres and Weiss, and below which this can fail. This provides a range in which $\nu $ is singular with respect to the uniform product measure, but $\nu $ -almost every point is simply Poisson generic.

Abstract In this paper, we introduce a new technique to study the distribution in residue classes of sets of integers with digit and sum-of-digits restrictions. From our main theorem, we derive a necessary and sufficient condition for integers with missing digits to be uniformly distributed in arithmetic progressions, extending previous results going back to the work of Erdős, Mauduit and Sárközy. Our approach uses Markov chains and does not rely on Fourier analysis as many results of this nature do. Our results apply more generally to the class of multiplicatively invariant sets of integers. This class, defined by Glasscock, Moreira and Richter using symbolic dynamics, is an integer analogue to fractal sets and includes all missing digits sets. We address uniform distribution in this setting, partially answering an open question posed by the same authors.

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 $a,a+d,\ldots, a+kd,a+\Omega(d)\in A.$