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Consider the random circle rotation $x \to x + Z \text{ mod 1}$ on $([0, 1], \text{Lebesgue})$ where at each rotation, $Z$ is uniformly distributed on $[0, 1]$ and independent of previous rotations. I …

Let $(X, T, \mu)$ be an ergodic measure preserving system with finite measure, and $f \in L^{\infty} (X)$. Define the maximal orbit deviation function $D_f: X \to \mathbb R$ by $$D_f := \sup_{n, m \ge …

This is a direct follow up to For which rationals is this exponential sum bounded? Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$. What is the Hausdorff dimension of the …

Notation: Here $S^1$ denotes the circle, which we view as the unit sphere in $\mathbb C$. We equip the circle with its natural length metric. Let $\{\epsilon_n\}_{n \geq 1}$ be iid uniformly distribut …

In Furstenberg's proof of the multiple recurrence theorem in ergodic theory, one makes use of the concept of compact and weak mixing extensions of a measure preserving system. The following definition …

Let $\mathbf X := (X, \mathcal S, \mu, T)$ be an ergodic measure preserving system with finite measure such that for every increasing sequence $\{n_k\}$ of natural numbers whose natural density exists …

I have recently been interested in going deeper into ergodic theory, beyond an introductory level of knowledge. Background wise, my training has mostly been in stochastic analysis, and I have a cursor …

Let $0 < a < 1$ be an irrational number. Is it true that $$\liminf_{n \in \mathbb N, n \to \infty} n \{na\} = 0?$$ Note: Here $\{\cdot\}$ denotes the fractional part.

Consider the following continuous analogue of a card shuffling process: Let $Y_i, Z_i$ ($i \in \mathbb Z^+$) be sequences of jointly independent uniformly distributed random variables on $[0, 1]$. Den …

Let $(X, T, \mathcal F, \mu)$ be a nonatomic standard probability space equipped with a measure preserving transformation $T$. We say $T$ is uniformly weak mixing if for every $\varepsilon > 0$, there …

Let $(X, T, \mathcal F, \mu)$ be an ergodic measure preserving system with finite measure. Suppose $T$ is uniformly recurrent, in the following sense: For every $A \in \mathcal F$, there exists an $M …

Let $(\Omega, \mathcal F, \mu)$ be a standard probability space. Question: For each $f \in L^\infty (\Omega)$, does there exist an ergodic measure preserving transformation $T: \Omega \to \Omega$ such …

Let $S =[0, 1]^2$ denote the unit square in $\mathbb R^{2}$. For any subset $A$ of $S$ let $A^{c}$ denote its complement in $S$, and $\overline{A}$ its closure in $S$. Given a measurable map $g: W \t …

Note: This is a concrete case of the following question: Are almost all measure-preserving flows on compact manifolds ergodic? Let $M$ be a Riemannian manifold with its natural Riemannian measure, and …

Note: By upper/lower density, we shall mean the upper/lower asymptotic density as given here. Question: For any subset $S \subset \mathbb N$ with positive upper density, does there exists a $\varepsil …