Possible sizes of sumsets

Abstract: Nathanson introduced the range of cardinalities of $h$-fold sumsets $ \mathcal{R}(h,k):= \{|hA|:A \subseteq \mathbb{Z} \text{ and }|A| = k\}. $ Following a remark of Erdös and Szemerédi that determined the form of $\mathcal{R}(h,k)$ when $h=2$, Nathanson asked what the form of $\mathcal{R}(h,k)$ 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 $\mathcal{R}(h,k)$ is the entire interval $\left[hk-h+1,\binom{h+k-1}{h}\right]$ except for a specified set of $\binom{h-1}{2}$ numbers. Moreover, we show that one can take $k_3 = 2$.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)