Possible Sizes of Sumsets

TL;DR

This work resolves Nathanson’s question on the possible sizes of $h$-fold sumsets for large enough $k$: for fixed $h$, there exists $k_h$ such that $R(h,k)$ fills the integer interval $\left[hk-h+1,\binom{h+k-1}{h}\right]$ except for a specified triangular set $\Delta_{h,k}$ of size $\binom{h-1}{2}$. The authors prove the $h=3$ case exactly and provide a general framework for all $h$ via two complementary constructions: a quadratic-sized initial block from $(h,d)$-filling sets and a large tail built by merging dense and sparse components whose sumset sizes are controlled through generating functions and a hypercube IVT argument. The combination yields two main propositions: one establishing a substantial early interval outside $\Delta_{h,k}$ and another guaranteeing coverage up to the top endpoint for large $k$; together they imply the claimed structure for $\mathcal{R}(h,k)$ and confirm the conjectured asymptotic form with an explicit bound $k_3=2$. The paper also discusses conjectural refinements, restricted sumsets, and extensions to groups with torsion, outlining several promising directions for future work and potential improvements to the threshold $k_h'$.

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

Figures (4)

• Figure 1: We trivially have $\mathcal{R}(6,7) \subseteq [37,924]$. The gray triangle represents the numbers in $\Delta_{6,7}$ and Theorem \ref{['mainthm']} shows that there are no elements of $\mathcal{R}(6,7)$ in $\Delta_{6,7}$. Conjecture \ref{['mainconj']} says that $\mathcal{R}(6,7)$ is all of the listed numbers outside the gray triangle.
• Figure 2: The blue and red points are all of the vertices of $\mathcal{G}_{6}$; these are the possible values of $g(A) = (|2A|,|3A|)$ when $|A| = 6$. The red points indicate the path constructed in the proof of Theorem \ref{['thmh3']} to connect $(12,18)$ to $(21,56)$.
• Figure 3: An example of $A'$ from the proof of Lemma \ref{['lemmaauxilliary']} with $d = 100$.
• Figure 4: The points $(|3A|,|4A|)$ for all $A \subset \mathbb{Z}$ with $|A| = 6$. The colors correspond to the minimum value of $|2A|$ for specified $|3A|$ and $|4A|$.

Theorems & Definitions (49)

• Definition 1.1: nathanson25Problems
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Conjecture 1.6
• Theorem 1.7
• Definition 2.1
• Lemma 2.2
• proof