Structure in additively nonsmoothing sets

Sets with many additive quadruples are guaranteed to have many additive octuples, by H"{o}lder’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 $F_3^N$.