Possible Sizes of Sumsets

Abstract

Nathanson introduced the range of cardinalities of $h$-fold sumsets $R(h,k):=\{|hA|:A\subset \mathbb{Z}\text{ and }|A|=k\}.$ Following a remark of Erdős and Szemerédi that determined the form of $R(h,k)$ when $h=2$, Nathanson asked what the form of $R(h,k)$ is for arbitrary $h,k\in \mathbb{N}$. For $h\in \mathbb{N}$, we prove there is some constant $k_h\in \mathbb{N}$ such that if $k>k_h$, then $R(h,k)$ is the entire interval $\left[hk-h+1,\left(\genfrac{}{}{0ex}{}{h+k-1}{h}\right)\right]$ except for a specified set of $\left(\genfrac{}{}{0ex}{}{h-1}{2}\right)$ numbers. Moreover, we show that one can take $k_3=2$.

arXiv comment17 pages, 4 figures, version published by Discrete Analysis