Relative sizes of iterated sumsets
Noah Kravitz
TL;DR
This work resolves Nathanson's qualitative questions on the relative growth of iterated sumsets by constructing, for any prescribed $n$ and $H$, finite sets $A_1, obreak[0] obreak[0] obreak[0] A_n$ in a sufficiently large abelian group $G$ so that, for each $h=1,\dots,H$, the sizes $|hA_1|,\dots,|hA_n|$ realize any given relative order specified by permutations $\sigma_h \in \mathfrak{S}_n$. The authors develop a flexible, multi-block framework: for each $h$ they build dominating blocks $B_{h,i}$ and then form the $A_k$ as Cartesian products of these blocks to realize the desired orderings. The reduction to two model settings, $(\mathbb{Z}/p\mathbb{Z})^{N_p}$ and $\mathbb{Z}$, together with constructive building blocks (in positive characteristic via $X(s,t)$-type sets and in characteristic zero via $Y(u,v)$-type sets, later enhanced by $Z(u,v,w)$), yields a complete proof and extends to prescribing equalities and limiting orders in the integers. An alternative, purely integer-based construction using multi-scale unions of progressions is also developed, offering complementary insights and highlighting distinct underlying mechanisms. Overall, the paper advances our understanding of how relative sumset growth can be engineered across several values of $h$ and demonstrates both broad applicability and deep structure behind iterated sumsets.
Abstract
Let $hA$ denote the $h$-fold sumset of a subset $A$ of an abelian group. Resolving a problem of Nathanson, we show that for any prescribed permutations $σ_1, \ldots, σ_H \in \mathfrak{S}_n$, there exist finite subsets $A_1, \ldots, A_n \subseteq \mathbb{Z}$ such that for each $1 \leq h \leq H$, the relative order of the quantities $|h A_1|, \ldots, |h A_n|$ is given by $σ_h$. We also establish extensions where $\mathbb{Z}$ is replaced by any other infinite abelian group or where one prescribes some equalities (not only inequalities) among the sumset sizes.