(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.