Structure in additively nonsmoothing sets

Sets with many additive quadruples are guaranteed to have many additive octuples, by Hölder's inequality. Sets with not many more than this are said to be additively nonsmoothing. We give a new proof of a structural theorem for nonsmoothing sets that originally appeared in work of the authors (\cite{BK}) on the size of cap sets in $F_3 ^N$.