Lacunary Counterexamples to a Distinct-Summand Freiman Container Problem

Fix \(K>1\) . We construct finite nonempty sets \(A,B\subseteq\mathbb Z\) with \(|A|>|B|\) and \(|A+B|<K|A|\) for which \(B\) is not contained in any generalized arithmetic progression of bounded rank and size \(O_K(|A|)\) . More strongly, for every prescribed rank bound \(d\) and size constant \(C\) , one counterexample defeats all progressions of rank at most \(d\) and cardinality at most \(C|A|\) . The construction is \(B=\{0,1,N,\ldots,N^{r-1}\}\) and \(A=kB\) . Uniqueness of base- \(N\) digits gives exact polynomial growth \(|sB|=\binom{s+r}{r}\) for \(s<N\) , whereas every rank- \(d\) generalized arithmetic progression has \(h\) -fold growth at most \(h^d\) . This answers Problem 2.5 from the 2004 AIM list on recent trends in additive combinatorics in the negative for every \(K>1\) .