(as opposed to the containment going the other way). Also, just above this, it may be more clear to write let be arbitary as opposed to let be arbitrary, to emphasize that one may take , but must be avoided to assure that the sets we wish to apply Lemma 3.18 to are nonempty.

*[Corrected, thanks - T.]*

*[One can for instance choose a random set where each of the are chosen independently and uniformly from if one wishes, but it is not necessary as one ends up with the same distribution of a random variable in either case. -T.]*

It seems that a better description might be: “Three decades later, Bourgain used….”

]]>*[Corrected, thanks - T.]*