12.1, 12.2

bibliography/references.bib

@@ -46,7 +46,7 @@ year = {2012},
 }
 @MISC{3722713,
     TITLE = {Embedding of countable linear orders into $\Bbb Q$ as topological spaces},
-    AUTHOR = {Eric Wofsey (https://math.stackexchange.com/users/86856/eric-wofsey)},
+    AUTHOR = {Eric Wofsey},
     HOWPUBLISHED = {Mathematics Stack Exchange},
     NOTE = {URL:https://math.stackexchange.com/q/3722713 (version: 2020-06-16)},
     EPRINT = {https://math.stackexchange.com/q/3722713},

inputs/lecture_16.tex

@@ -156,6 +156,7 @@ By Zorn's lemma, this will follow from
 \end{theorem}
 \yaref{thm:furstenberg} allows us to talk about ranks of distal minimal flows:
 \begin{definition}[{\cite[{}13.1]{Furstenberg}}]
+    \label{def:floworder}
     Let $(X,T)$ be a quasi-isometric flow,
     and let $\eta$ be the smallest ordinal
     such that there exists a quasi-isometric system $\{(X_\xi, T), \xi \le \eta\}$

inputs/lecture_19.tex

@@ -272,9 +272,6 @@ More generally we can show:
     In particular,
     $(X'_{\xi+1}, T)$ is a factor of $(X_{\xi+1}, T)$.
 \end{proof}
-
-% TODO ANKI-MARKER
-
 \begin{example}[{\cite[p. 513]{Furstenberg}}]
     \label{ex:19:inftorus}
     Let $X$ be the infinite torus

inputs/lecture_20.tex

@@ -19,6 +19,10 @@
     Here I'll try to only use multiplicative notation.
 \end{remark}
 }{}
+
+
+% TODO ANKI-MARKER
+
 We will be studying projections to the first $d$ coordinates,
 i.e.
 \[
@@ -34,9 +38,9 @@ For $d = 1$ we get the circle rotation $x \mapsto e^{\i \alpha} x$.
     \[
     H_m \coloneqq  \{x \in S^1 : x^m = 0\}
     \]
-    for some $m \in \Z$.
+    for some $m \in \Z$.%
+    \footnote{cf.~\yaref{s12e2}}
 \end{fact}
-\todo{Homework!}
 We will show that $\tau_d$ is minimal for all $d$,
 i.e.~every orbit is dense.
 From this it will follow that $\tau$ is minimal.
@@ -45,7 +49,6 @@ Let $\pi_n\colon X \to  (S^1)^n$ be the projection to the first $n$
 coordinates.
 
 
-
 \begin{lemma}
     \label{lem:lec20:1}
     Let $x,x' \in X$ with $\pi_n(x) = \pi_n(x')$

inputs/tutorial_14.tex

@@ -4,9 +4,58 @@
 \nr 1
 % Examinable
 
-% TODO  (there is a more direct way to do it, not using analytic / coanalytic)
+Let $\LO(\N) \overset{\text{closed}}{\subseteq} 2^{\N\times \N}$ denote the set of linear orders on $\N$.
+
+Let $S \subseteq \LO(\N)$ be the set of orders having a least
+element and such that every element has an immediate successor.
+\begin{itemize}
+    \item $S$ is Borel in $\LO(\N)$:
+
+        Let $M_n \subseteq \LO(\N)$ be the set of orders with minimal element $n$.
+        Let $I_{n,m} \subseteq \LO(\N)$ be the set of orders such
+        that $m$ is the immediate successor of $n$.
+
+        Clearly $S = \left(\bigcap_n \bigcup_{m\neq n} I_{n,m}\right) \cap \bigcup_n M_n$,
+        so it suffices to show that $M_n$ and $I_{n,m}$ are Borel.
+        It is $M_n = \bigcap_{m\neq n} \{\prec : m \not\prec n\}$
+        and $I_{n,m} = \{\prec: n \prec m\} \cap \bigcap_{i} \{\prec : n \preceq i \preceq m \implies n = i \lor n = m \}$.
+    \item Give an example of an element of  $S$ which is not well-ordered:
+
+        Consider $\{1 - \frac{1}{n} : n \in \N^+\} \cup \{1 +   \frac{1}{n} : n \in \N^{+}\} \subseteq \R$
+        with the order $<_\R$.
+        This is an element of $S$,
+        but $\{x \in S: x \ge 1\}$ has no minimal element,
+        hence it is not well-ordered.
+
+\end{itemize}
+
 \nr 2
 % Examinable
+Recall the definition of the circle shift flow $(\R / \Z, \Z)$
+with parameter $\alpha \in \R$, $1 \cdot x \coloneqq  x + \alpha$.
+
+\begin{itemize}
+    \item If $\alpha \not\in  \Q$, then $(\R / \Z, \Z)$ is minimal:
+
+        This is known as \href{https://en.wikipedia.org/wiki/Dirichlet's_approximation_theorem}{Dirichlet's Approximation Theorem}.
+
+    \item Consider $\R/\Z$ as a topological group.
+        Any subgroup $H$ of $\R / \Z$ is dense in $\R / \Z$
+        or of the form $H = \{ x \in \R / \Z | mx = 0\}$
+        for some $m \in \Z$.
+
+
+        If $H$ contains an irrational element $\alpha$, then
+        it is dense by the previous point.
+
+        Suppose that $H \subseteq \Q / \Z$.
+        Let $D$ be the set of denominators of elements of $H$
+        written as irreducible fractions.
+        If $D$ is finite,
+        then  $H = \{x \in \R / \Z : \mathop{lcm}(D)x = 0\}$.
+        Otherwise $H$ is dense, as it contains
+        elements of arbitrarily large denominator.
+\end{itemize}
 
 
 \nr 3