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On the Sumset of Sets of Size $k$
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abstract
The set $\mathcal{R}_{G}(h,k)$ consists of all possible sizes for the $h$-fold sumset of sets containing $k$ elements from an additive abelian group $G$. The exact makeup of this set is still unknown, but there has been progress towards determining which integers are present. We know that $\mathcal{R}_{G}(h,k)\subseteq\left[hk-h+1,\binom{h+k-1}{h}\right]$, where the right side is an interval of integers that includes the endpoints. These endpoints are known to be attained. We will prove that the integers in $\left[hk-h+2,hk-1\right]$ are not possible sizes for the $h$-fold sumset of a set containing $k\geq 4$ elements of a torsion-free additive abelian group $G$. Furthermore, we will confirm that this interval can't be made larger by exhibiting a subset of $G$ whose $h$-fold sumset has size $hk$.
Forward citations
Cited by 3 Pith papers
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Possible Sizes of Sumsets
For fixed h and large k, the possible sizes of h-fold sumsets of k-element integer sets form the full interval [hk−h+1, C(h+k−1,h)] minus C(h−1,2) specified numbers; the h=3 case is settled for all k>2.
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Additive sumset sizes with tetrahedral differences
For each h and i0 from 0 to h-1, the set {0,1,h+1,(h+1-i0)(h+1)} has h-fold sumset size binomial(h+3,3) minus binomial(i0+2,3).
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Triangular and tetrahedral number differences of sumset sizes in additive number theory
For 4-element sets of integers, the most popular h-fold sumset sizes appear to equal C(h+3,3) minus the first h tetrahedral numbers, but only computer experiments are given.
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