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Optimally small sumsets in groups. I: The supersmall sumsets property, the \(\mu_G^{(k)}\) and the \(\nu_G^{(k)}\) functions. (English) Zbl 1131.11014

Let \(G\) be an Abelian group. We are interested in subsets \(\mathcal{A}_1, \dots, \mathcal{A}_k \subset G\) such that the cardinality of the set of sums \(\mathcal{A}_1 + \dots + \mathcal{A}_k\) is as small as possible. More precisely, suppose that natural numbers \(r_1, \dots, r_k\) are given and define
\[ \mu^{(k)}_G(r_1, \dots, r_k) := \min\biggl\{\Bigl| \sum_i\mathcal{A}_i\Bigr|: \mathcal{A}_i \subset G, | \mathcal{A}_i| = r_i\;(i=1,\dots, k) \biggr\}. \]
One of the main results is a formula for \(\mu^{(k)}_{G}\) for arbitrary Abelian groups \(G\) (not necessarily finite):
\[ \mu^{(k)}_G(r_1, \dots, r_k) = \min\left\{ d\cdot\left(\sum_i\left\lceil\frac{r_i}{d}\right\rceil - k + 1\right) : d \text{ is the order of an }a\in G \right\}. \]
Indeed, this is what Kneser’s theorem would suggest.
On his way to a proof, the author introduces the “generalized supersmall sumset property”. A group has this property if for any choice \(1 \leq r_1, \dots, r_k \leq | G| \), subsets \(\mathcal{A}_1, \dots, \mathcal{A}_k \subset G\) exist such that \(\sum_i\mathcal{A}_i\) is as small as one could expect in a suitable (slightly technical) sense. The author proves this property not only for arbitrary Abelian groups, but also for solvable ones.
Finally, he considers a variant \(\nu^{(k)}_G(r_1, \dots, r_k)\) of \(\mu^{(k)}_G(r_1, \dots, r_k)\) where one requires that the sumset \(\sum_i\mathcal{A}_i\) contains an element which has only one representation as a sum of elements of the \(A_i\). He computes this function in some cases and proves good lower bounds in general.
For Part II, se the following review Zbl 1131.11015.

MSC:

11B75 Other combinatorial number theory
11P99 Additive number theory; partitions
20F16 Solvable groups, supersolvable groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20K01 Finite abelian groups

Citations:

Zbl 1131.11015
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