I have a few questions on Section 8.2 (Sanders-Croot-Sisask-theory), related to remark 8.2.3.

– p.180: exercise 8.2.1. S^m is contained in A^2A^{-2}=A^4 if A is assumed either symmetric, or an approximate subgroup, but in remark 8.2.3, you mention that small doubling+finite+non-empty suffices. But in that case, we can’t get S^m contained in A^4, can we?

– p.180: exercise 8.2.2. If f is supported on A^2, Sf, Tf would be supported on A^3, in which case the small doubling assumption is not quite enough to guarantee the variance bound (we would need small tripling in full generality, of course if A is assumed to be an approximate group, that should be ok). It seemed that it is important that f be supported on gA (which is the case if f=1_{gA}), since then the support of Tf would be of size |A^2|<K|A|.

– p.181: you say that |A^2|/M 1_{y_iA} has an l^2-norm of O_K(|A|^3/2), but I found a bound depending on both K and M.

Is there a natural way that one can, without any symmetry assumption on A, have A^2A^{-2} contained in some higher iterated power A^k?

Thanks!

-L

P. 5, l. 5 in the proof:

The ‘s in shouldn’t be in italic mode.

P. 6, l. 2 in exercise 1.1.4:

The ‘s in shouldn’t be in italic mode.

P. 8, l. 1 after the two times two matrix:

Is the ‘s in supposed to be in italic mode?

P. 17, footnote 4:

Use \dots instead of “…” and remember a comma after the definition of the group G_{3}.

In general:

When typing a map, use \colon instead of a normal colon. (The spacing is incorrect.) Also, use \coloneqq from the mathtools package instead of “:=”.

*[Thanks, this will be incorporated into the next revision of the ms. -T.]*