LemmaA \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. □
PropA\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
(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
LemmaA \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. □
PropositionA\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
[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
Notes
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.
