Peter Krautzberger on the web

Red workbook, p8


red workbook, p8
Red Workbook, p.8


  • Lemma $A \subseteq S$ (a) $A \in p \in K(\beta S)$ (also pws) => $\exists g \subseteq_ e S: \stackrel{\stackrel{[unreadable]}{\downarrow}}{\beta S \cdot p} \subseteq \widehat{ \bigcup_ {x \in g} x^{-1} A }$
    • (b) $T := \bigcup_ {x \in g} x^{-1} A$ dick, $g \subseteq_ e S$ (also $A$ pws)
      • => $\beta S \cdot q \subseteq \widehat{T}$ fuer ein $q \in \beta S$
      • => $A \in x \cdot q$ fuer ein $q\in \beta S$, ein $x\in g$
  • Beweis
    • (b): $q \in K(\beta S)$ mit $\beta S \cdot q \subseteq \widehat{T}$ (Prop)
      • $q \in \beta S \cdot q \subseteq \widehat{T} \Rightarrow \exists x \in g: x^{-1}A \in q \Rightarrow x \cdot q \in \widehat{A}$
    • (a) $L := \beta S \cdot p$ min. LID , $p \in L$.
      • Beh: $L \subseteq \bigcup_ {t \in S} \widehat{t^{-1}A }$ [ $x \in L, p \in \beta S \cdot x = cl_ {\beta S} (s \cdot x)$
        • da $p\in \widehat{A}$ offen => $\exists t \in S: t\cdot x \in \widehat{A} \Rightarrow x \in \widehat{t^{-1}A}$]
      • $L$ kompakt, also $\underset{\stackrel{=}{\beta S \cdot p} }{L} \subseteq \bigcup_ {t \in g} \widehat{ t^{-1}A}$ fuer ein $g \subseteq_ e S$. □
  • Prop $A\subseteq S$; AeQ: (a) $A$ pws (b) $\exists y \in S: y^{-1}A$ zentral
    • (c) $\exists D \subseteq S \text{ synd} \forall d \in D: d^{-1}A$ zentral
  • Beweis
    • (b) => (a): nimm $\epsilon \in E_ \min(\beta S): y^{-1}A \in \epsilon \Rightarrow A \in y \cdot \epsilon \in K(\beta S)$
    • (a) => (c): Sei $e \subseteq_ e S$ mit $\bigcup_ {t \in e} t^{-1}A = T$ dick;
      • sei $L\subseteq \widehat{T}$ min LID, sei $\epsilon \in L \cap E(\beta S)$
      • Sei $y\in \epsilon$ mit $\epsilon \in \widehat{ y^{-1} A }$. Im DS $\beta S$:
      • $\epsilon$ uniform rekurrent, $B = R(\epsilon, \widehat{ y^{-1}}A)$ syndetisch
      • $D: y \cdot B$ (syndetisch! (Bem.)) und
      • $\forall d\in D: d^{-1}A \in \epsilon$ (also zentral) [$d = y\cdot b$ fuer ein $b \in B$, $b\epsilon \in y^{-1}A$
        • $y b \epsilon \in \widehat{A}, d\epsilon \in \widehat{A}, d^{-1}A \in \epsilon$]

partial Translation

  • Lemma $A \subseteq S$
    • (a) $A \in p \in K(\beta S)$ (hence piecewise syndetic (pws)) => $\exists g \subseteq_ e S: \beta S \cdot p \subseteq \widehat{ \bigcup_ {x \in g} x^{-1} A }$ [[so this union is thick!]]
    • (b) $T := \bigcup_ {x \in g} x^{-1} A$ thick, $g \subseteq S$ finite (so $A$ is pws)
      • => $\beta S \cdot q \subseteq \widehat{T}$ for some $q \in \beta S$
      • => $A \in x \cdot q$ for some $q\in \beta S$ and some $x\in g$
  • Proof
    • (b): $q \in K(\beta S)$ with $\beta S \cdot q \subseteq \widehat{T}$ (by the previous proposition) [[workbook p 7]]
      • $q \in \beta S \cdot q \subseteq \widehat{T} \Rightarrow \exists x \in g: x^{-1}A \in q \Rightarrow x \cdot q \in \widehat{A}$
    • (a) $L := \beta S \cdot p$ minimal left ideal (LID) , $p \in L$.
      • Claim: $L \subseteq \bigcup_ {t \in S} \widehat{t^{-1}A }$
        • $x \in L, p \in \beta S \cdot x = cl_ {\beta S} (s \cdot x)$ since $p\in \widehat{A}$ open => $\exists t \in S: t\cdot x \in \widehat{A} \Rightarrow x \in \widehat{t^{-1}A}$]
      • $L$ compact, hence $\underset{\stackrel{=}{\beta S \cdot p} }{L} \subseteq \bigcup_ {t \in g} \widehat{ t^{-1}A}$ for some finite $g \subseteq S$. □
  • Proposition $A\subseteq S$; TFAE
    • (a) $A$ pws
    • (b) $\exists y \in S: y^{-1}A$ central
    • (c) $\exists D \subseteq S \text{ syndetic} \forall d \in D: d^{-1}A$ central
  • Proof
    • (b) => (a): take $\epsilon \in E_ \min(\beta S): y^{-1}A \in \epsilon \Rightarrow A \in y \cdot \epsilon \in K(\beta S)$
    • (a) => (c): Let $e \subseteq S$ finite with $\bigcup_ {t \in e} t^{-1}A = T$ thick;
      • Let $L\subseteq \widehat{T}$ min. LID, let $\epsilon \in L \cap E(\beta S)$
      • Let $y\in \epsilon$ with $\epsilon \in \widehat{ y^{-1} A }$. In the dynamical system $\beta S$:
      • $\epsilon$ is uniformly recurrent, $B = R(\epsilon, \widehat{ y^{-1}}A)$ syndetic
      • $D: y \cdot B$ (syndetic! (by the above remark.)) undand
      • $\forall d\in D: d^{-1}A \in \epsilon$ (hence central)
        • [proof]: $d = y\cdot b$ for some $b \in B$, $b\epsilon \in y^{-1}A$, $y b \epsilon \in \widehat{A}, d\epsilon \in \widehat{A}, d^{-1}A \in \epsilon$


More wonderful stuff about thick, piecewise syndetics, and central sets.

The lemma tells us that pws could be called “almost thick” – a finite set of translations is enough to make a pws set thick. The proposition on the other hand tells us that pws is surprisingly close to being central – just one translation! (just keep in mind they are very much not the same notion). In addition, such a translation happens very, very frequently (a syndetic set!).

Somehow, I find this to be a lot of fun even if it’s not particularly surprising – minimal idempotent ultrafilters are just so incredibly rich.