Peter Krautzberger · on the web

# Red workbook, p8

### Transcript

• Lemma (a) (also pws) =>
• (b) dick, (also pws)
• => fuer ein
• => fuer ein , ein
• Beweis
• (b): mit (Prop)
• (a) min. LID , .
• Beh: [
• da offen => ]
• kompakt, also fuer ein . □
• Prop ; AeQ: (a) pws (b) zentral
• (c) zentral
• Beweis
• (b) => (a): nimm
• (a) => (c): Sei mit dick;
• sei min LID, sei
• Sei mit . Im DS :
• uniform rekurrent, syndetisch
• (syndetisch! (Bem.)) und
• (also zentral) [ fuer ein ,
• ]

### partial Translation

• Lemma
• (a) (hence piecewise syndetic (pws)) => [[so this union is thick!]]
• (b) thick, finite (so is pws)
• => for some
• => for some and some
• Proof
• (b): with (by the previous proposition) [[workbook p 7]]
• (a) minimal left ideal (LID) , .
• Claim:
• since open => ]
• compact, hence for some finite . □
• Proposition ; TFAE
• (a) pws
• (b) central
• (c) central
• Proof
• (b) => (a): take
• (a) => (c): Let finite with thick;
• Let min. LID, let
• Let with . In the dynamical system :
• is uniformly recurrent, syndetic
• (syndetic! (by the above remark.)) undand
• (hence central)
• [proof]: for some , ,

### 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.