The size of the h-fold sumset is the binomial maximum until h reaches a lattice minimum, then drops by a second binomial term until the next minimum, producing triangular-number differences for four-element sets.
Inverse problems for sumset sizes of finite sets of integers
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Let $A$ be a finite set of integers and let $hA$ be its $h$-fold sumset. This paper investigates the sequence of sumset sizes $( |hA| )_{h=1}^{\infty}$, the relations between these sequences for affinely inequivalent sets $A$ and $B$, and the comparative growth rates and configurations of the sumset size sequences $( |hA| )_{h=1}^{\infty}$ and $( |hA| )_{h=1}^{\infty}$.
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On Nathanson's Triangular Number Phenomenon
The size of the h-fold sumset is the binomial maximum until h reaches a lattice minimum, then drops by a second binomial term until the next minimum, producing triangular-number differences for four-element sets.