\[
\newcommand{\C}{\mathbb{C}}
\newcommand{\haar}{\mathsf{m}}
\newcommand{\B}{\mathscr{B}}
\newcommand{\P}{\mathcal{P}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\g}{>}
\newcommand{\l}{<}
\newcommand{\intd}{\,\mathsf{d}}
\newcommand{\Re}{\mathsf{Re}}
\newcommand{\area}{\mathop{\mathsf{Area}}}
\newcommand{\met}{\mathop{\mathsf{d}}}
\newcommand{\emptyset}{\varnothing}
\DeclareMathOperator{\borel}{\mathsf{Bor}}
\newcommand{\symdiff}{\mathop\triangle}
\DeclareMathOperator{\leb}{\mathsf{Leb}}
\DeclareMathOperator{\cont}{\mathsf{C}}
\DeclareMathOperator{\lpell}{\mathsf{L}}
\newcommand{\lp}[1]{\lpell^{\!\mathsf{#1}}}
\]
Week 5 Worksheet
- Take $X = \{0,1\}^\N$. Equip $X$ with the metric
\[
\met(x,y) = \sum_{n=1}^\infty \dfrac{x(n) - y(n)}{2^n}
\]
and define $(Tx)(n) = x(n+1)$ for all $n \in \N$ and all $x \in X$.
- Write down a point $x \in X$ whose $T$ orbit is finite.
- Write down a point $x \in X$ whose $T$ orbit is not dense.
- Write down a point $x \in X$ whose $T$ orbit is dense.
- Put $X = [0,1) \times [0,1)$ and define $T : X \to X$ by
\[
T(x,y) = (x+\alpha \bmod 1, y+x+\alpha \bmod 1)
\]
for all $(x,y) \in X$.
- Verify that
\[
T^n(x,y) = (x+n\alpha, y + nx + \tbinom{n}{2} \alpha)
\]
for all $n \in \N$ and all $x,y \in X$.
- When is the orbit of $(0,0)$ dense in $[0,1)^2$?
- Fix a bounded sequence $x_n$ in $\R$. Suppose
\[
\gamma = \lim_{N \to \infty} \dfrac{1}{N} \sum_{n=1}^N x_n
\]
exists. Prove that
\[
\lim_{N \to \infty} \dfrac{1}{N} \sum_{n=k}^{N+k} x_n = \gamma
\]
for each $k \in \N$.
- Is the sequence $n \mapsto \log n \bmod 1$ uniformly distributed?
- Put $X = [0,1) \times [0,1)$.
- Define what it means for a sequence $n \mapsto x_n$ in $X$ to be uniformly distributed in $X$.
- When is
\[
n \mapsto (n \alpha \bmod 1, n \alpha \bmod 1)
\]
uniformly distributed in $X$?
-
Given a Borel measure $\mu$ on $X = [0,1)$ with $\mu(X) = 1$ and a strictly increasing sequence $N \mapsto r(N)$ in $\N$ say that a sequence $n \mapsto x_n$ in $X$ is $(r,\mu)$ uniformly distributed with respect to $\mu$ if
\[
\lim_{N \to \infty} \dfrac{1}{r(N)} \sum_{n=1}^{r(N)} f(x_n) = \int f \intd \mu
\]
for all continuous function $f : X \to \R$.
- Using the Riesz representation theorem, prove that for every sequence $n \mapsto x(n)$ in $X$ there is a a Borel measure $\mu$ on $X$ and a strictly increasing sequence $N \mapsto r(N)$ in $\N$ such that $x$ is $(r,\mu)$ uniformly distributed.
- Can a sequence $n \mapsto x_n$ in $X$ be $(r,\mu)$ and $(s,\nu)$ uniformly distributed with $\mu \ne \nu$?
- Say that $E \subset \N$ is syndetic if there is $r(1),\dots,r(k)$ with
\[
\N = (E - r(1)) \cup \cdots \cup (E - r(k))
\]
where $E - t = \{ s \in \N : t + s \in E \}$. Prove that
\[
\{ n \in \N : n \alpha \bmod 1 \in (\tfrac{1}{4},\tfrac{3}{4}) \}
\]
is syndetic for all irrational $\alpha$.