Optimally small sumsets in groups. III: The generalized increasingly small sumsets property and the \(v^{(k)}_G\) functions (Q2479901)

This is the third article in a series about subsets \({\mathcal A}_i\) of an Abelian group \(G\) such that the cardinality of the set of sums \({\mathcal A}_1 + \dots + {\mathcal A}_k\) is as small as possible. For fixed cardinalities \(r_i = |{\mathcal A}_i| \), define \(\mu^{(k)}_G(r_1, \dots, r_k)\) to be the minimal cardinality of the sumset. This function has already been computed in the first article of the series [\textit{A. Plagne}, Unif. Distrib. Theory 1, No. 1, 27--44 (2006; Zbl 1131.11014)]. In the present article, we additionally require that the sumset contains at least one element which has a unique representation as sum \(a_1 + \dots + a_k\), \(a_i \in \mathcal{A}_i\). Define \(\nu^{(k)}_G(r_1, \dots, r_k)\) to be the minimal cardinality of such sumsets (or \(\infty\) if no such set exists). A lower bound for \(\nu^{(k)}_G\) has already been given in the first article of the series: \(\nu^{(k)}_G(r_1, \dots, r_k) \geq r_1 + \dots + r_k - k + 1\). This lower bound is attained e.g. if \(G\) contains a cyclic subgroup of cardinality at least \(r_1 + \dots + r_k - k + 1\). As an easy corollary from previous results, the author shows that moreover this bound is attained whenever \(\mu^{(k)}_G(r_1, \dots, r_k) = r_1 + \dots + r_k - k + 1\). However, in general \(\nu^{(k)}_G\) might be larger. The main goal of this article is to develop tools to find upper bounds for \(\nu^{(k)}_G\). A key ingredient is the ``generalized increasingly small sumsets property''. Roughly, a group \(G\) has this property if for any \(r_i \leq | G| \), sets \(\mathcal{A}_i\) of cardinality \(r_i\) can be found such that the sumset is sufficiently small; moreover, a certain flexibility in choosing these sets is required. The author proves this property for arbitrary Abelian groups and then deduces an upper bound for \(\nu^{(k)}_G\) in the case \(k = 2\), as a function of \(\mu^{(k)}_G\). This upper bound is attained for example in the case \(\nu^{(2)}_{\mathbb{Z}/p\mathbb{Z}}(p,p) = 2p\).