Possible Sizes of Sumsets

TL;DR

This paper characterizes the possible sizes of h-fold sumsets for large sets, showing they form nearly complete intervals with few exceptions, extending known results from the case h=2 to general h.

Contribution

It proves that for large enough set sizes, the range of sumset cardinalities is almost continuous, with explicit bounds and a specific exception set, generalizing previous work.

Findings

For large k, R(h,k) covers nearly all integers in a specific interval.

The set of exceptions has size inom{h-1}{2}.

Explicitly, for h=3, the minimal k is 2.

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 Erd\H{o}s and Szemer\'edi 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 $\left[hk-h+1,\binom{h+k-1}{h}\right]$ except for a specified set of $\binom{h-1}{2}$ numbers. Moreover, we show that one can take $k_3 = 2$.