I guess in the beginning the link to “previous lecture notes” should be pointing to one lecture earlier (i.e. Lecture 8 not 9.)

*[Corrected, thanks – T.]*

I think you’re still missing the $\frac{1}{2}$.

Specifically, just above the “parallelogram law” link, it should be

$ET^b v = \frac{1}{2} (ET^a v + T^g ET^a v)$,

since the size of the multiset has now doubled.

]]>Ah, yes. There is a 1/2 missing in the RHS of E_{b \in A + {0,g}} = E_a T^a v + T^g E_a T^a v

*[Corrected, thanks – T.]*

Hmm you’re right; uncountable abelian groups don’t have Folner sequences, but they still have invariant means. I guess what I meant to say here is that one can establish the ergodic theorem and construct the Gowers-Host-Kra seminorms without explicit use of Folner sequences, even if the ambient group is still technically amenable.

]]>As far as I am aware all abelian groups are amenable (see e.g. Chapter 10 in “The Banach-Tarski paradox” by Stan Wagon). If I am not mistaken a slight modification of the procedure described in Theorem 2 should provide an invariant mean on bounded functions on the group.

]]>*[Corrected, thanks – T.]*

In the proof of Thm 2, there is a lim_{A\to H} (“we conclude that the limit…”)

I’m having some trouble seeing the Cauchy estimate from the parallelogram law in Thm 2. Taking X = ET^a v we have ||X – T^g X||^2 + ||X+ T^g X||^2 = 4||X||^2 and from ||X + T^gX||^2 \geq ||X||^2 – \epsilon we get ||X – T^g X||^2\leq \epsilon + 3||X||^2 instead of 4\epsilon

In the discussion immediately preceding Thm 3, E(f, X^G) has a \mathcal{E} instead of \mathbf

*[Corrected, thanks. The lower bound should be on , not . -T.]*