* [8]: “93-128” –> “93–128”

* [9]: The title should be in italic

* [10]: “261-264” –> “261–264”

* [16]: “327,1–29” –> “327, 1–29”

* [23]: The URL should be put in typewriter font, i.e., \href{https://arxiv.org/abs/1205.1330}{\texttt{https://arxiv.org/abs/1205.1330}}.

* [26]: “2009. 180–204” –> “2009, 180–204”

* [32]: “AMS 1994” –> “AMS, 1994”

* [33]: “R.A.” –> ” R. A.”

* [40]: The title should be in italic

* [44]: “Hungar.56” –> “Hungar. 56”

* [44]: “no. 1-2” –> “no. 1–2”

* [46]: Should “2004 El Escorial conference proceedings” be in italic?

*[Thanks, this will be corrected in the next version of the ms. -T.]*

It’s a long story, and I’ve actually forgotten parts of it at this point, but basically the first version of our argument (based on density increment methods) turned out to have some significant technical difficulties, so we put it aside for a few years until we found a different argument (based on energy increment methods) which worked; however the write up of this paper was extremely technical and we did not feel it was readable enough to be in a publishable state (though we did share the preprint with some close colleagues who requested it), so the paper again languished. About a year ago we were contacted to submit an article for a special issue of Mathematika honouring the work of Klaus Roth, and this encouraged us to rewrite the (quite lengthy) paper to a state that we were both happy with for submission, but we were only able to do so after we were both at MSRI earlier this year, and could devote a significant portion of time to the task.

]]>There are two obstacles to this currently. The first is that one would need a quantitative (local) inverse U^4 theorem of comparable strength to the quantitative (local) inverse U^3 theorem that is used in this paper, and at present no such quantitative theorem is known. Secondly (and more seriously), the application of Cauchy-Schwarz inequality that lower bounds (2) is not available for the analogue to (2) in the k=5 setting (which is a somewhat complicated quintilinear integral expression on a certain nilmanifold); indeed there is a counterexample by Ruzsa in the appendix to the Bergelson-Host-Kra paper mentioned in the post that suggests that no analogue of the Khintchine-type recurrence theorems proven here in the k=4 case will be available for k equal to 5 or higher.

]]>breakthrough was made by in 1998 by Gowers”; but should probably say, “A major breakthrough was made by Gowers in 1998”?

*[Thanks, this will be corrected in the next version of the ms. -T.]*

*[ See https://en.wikipedia.org/wiki/Time_complexity#Polylogarithmic_time -T.]*

I meant that (since the property of such subset is translation-invariant) it is possible that there are several maximal subsets with this property without any largest one (on which the above definition of is based), so in the above definition of it seems clearer to replace “cardinality of the largest subset ” by “largest cardinality of a subset ” (as in the definition of in the abstract of your paper.)

*[Wording changed – T.]*