Week 5 Worksheet

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1. Take $X=\{0,1{\}}^{\mathbb{N}}$. Equip $X$ with the metric $$\mathsf{d}(x,y)=\sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{x(n)-y(n)}{2^n}}$$ and define $(Tx)(n)=x(n+1)$ for all $n\in \mathbb{N}$ and all $x\in X$.
   1. Write down a point $x\in X$ whose $T$ orbit is finite.
   2. Write down a point $x\in X$ whose $T$ orbit is not dense.
   3. Write down a point $x\in X$ whose $T$ orbit is dense.
2. Put $X=[0,1)\times [0,1)$ and define $T:X\to X$ by $$T(x,y)=(x+\alpha mod1,y+x+\alpha mod1)$$ for all $(x,y)\in X$.
   1. Verify that $$T^n(x,y)=(x+n\alpha ,y+nx+(\genfrac{}{}{0ex}{}{n}{2})\alpha )$$ for all $n\in \mathbb{N}$ and all $x,y\in X$.
   2. When is the orbit of $(0,0)$ dense in $[0,1{)}^2$?
3. Fix a bounded sequence $x_n$ in $\mathbb{R}$. Suppose $$\gamma =\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=1}^N{x}_n$$ exists. Prove that $$\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{N}}\sum _{n=k}^{N+k}{x}_n=\gamma$$ for each $k\in \mathbb{N}$.
4. Is the sequence $n\mapsto \log nmod1$ uniformly distributed?
5. Put $X=[0,1)\times [0,1)$.
   1. Define what it means for a sequence $n\mapsto x_n$ in $X$ to be uniformly distributed in $X$.
   2. When is $$n\mapsto (n\alpha mod1,n\alpha mod1)$$ uniformly distributed in $X$?
6. Given a Borel measure $\mu$ on $X=[0,1)$ with $\mu (X)=1$ and a strictly increasing sequence $N\mapsto r(N)$ in $\mathbb{N}$ say that a sequence $n\mapsto x_n$ in $X$ is $(r,\mu )$ uniformly distributed with respect to $\mu$ if $$\underset{N\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{r(N)}}\sum _{n=1}^{r(N)}f(x_n)=\int f\mathsf{d}\mu$$ for all continuous function $f:X\to \mathbb{R}$.
   1. 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 $\mathbb{N}$ such that $x$ is $(r,\mu )$ uniformly distributed.
   2. Can a sequence $n\mapsto x_n$ in $X$ be $(r,\mu )$ and $(s,\nu )$ uniformly distributed with $\mu \ne \nu$?
7. Say that $E\subset \mathbb{N}$ is syndetic if there is $r(1),\dots ,r(k)$ with $$\mathbb{N}=(E-r(1))\cup \cdots \cup (E-r(k))$$ where $E-t=\{s\in \mathbb{N}:t+s\in E\}$. Prove that $$\{n\in \mathbb{N}:n\alpha mod1\in (\frac{1}{4},\frac{3}{4})\}$$ is syndetic for all irrational $\alpha$.