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Inverse problems for sumset sizes of finite sets of integers

Melvyn B. Nathanson

TL;DR

This work investigates inverse problems for the sequence of sumset sizes $|hA|$ of finite sets of integers, addressing how much the sequence determines the underlying set and how affine equivalence influences these sizes. It establishes that for large $h$, $|hA|$ is governed by a linear tail $|hA|=ha+1-N(A)-N(a-A)$, with recent refinements reducing the threshold to $h_0(A)=a-k+2$; it also proves that the size sequence does not determine affine equivalence, providing explicit counterexamples. The paper further demonstrates sumset-size oscillations by constructing pairs with matched early behavior but divergent later growth, and develops a $\tau$-type framework to study joint growth patterns across multiple sumsets, culminating in Kravitz’s theorem showing prescribed $\tau$-patterns are realizable in bounded coordinates. Overall, the results illuminate both the deterministic tail behavior and rich non-uniqueness and oscillation phenomena in sumset size growth, with extensions to multi-set and restricted-sum settings.

Abstract

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 $( |hA| )_{h=1}^{\infty}$.

Inverse problems for sumset sizes of finite sets of integers

TL;DR

This work investigates inverse problems for the sequence of sumset sizes of finite sets of integers, addressing how much the sequence determines the underlying set and how affine equivalence influences these sizes. It establishes that for large , is governed by a linear tail , with recent refinements reducing the threshold to ; it also proves that the size sequence does not determine affine equivalence, providing explicit counterexamples. The paper further demonstrates sumset-size oscillations by constructing pairs with matched early behavior but divergent later growth, and develops a -type framework to study joint growth patterns across multiple sumsets, culminating in Kravitz’s theorem showing prescribed -patterns are realizable in bounded coordinates. Overall, the results illuminate both the deterministic tail behavior and rich non-uniqueness and oscillation phenomena in sumset size growth, with extensions to multi-set and restricted-sum settings.

Abstract

Let be a finite set of integers and let be its -fold sumset. This paper investigates the sequence of sumset sizes , the relations between these sequences for affinely inequivalent sets and , and the comparative growth rates and configurations of the sumset size sequences and .

Paper Structure

This paper contains 8 sections, 9 theorems, 100 equations.

Table of Contents

  1. The sequence of sumset sizes
  2. The direct problem for sumset sizes
  3. Inverse problems for sumset size
  4. Sumset size oscillations
  5. Open problems
  6. $\tau$-type problems
  7. Added January 31, 2025
  8. Table of $|hA_{4,w}|$ for $A_{4,w} = [0,4] \cup \{w \}$, $h \in [2,9]$ and $w \in [5,25] \cup \{30,35, 40,45, 50\}$

Key Result

Theorem 1

Let $A$ be a finite set of $k \geq 2$ integers with $\min(A) = 0$, $\max(A) = a$, and $\gcd(A)=1$. There exist integers $C$ and $D$ and subsets $\mathcal{C} \subseteq [0,C-2]$ and $\mathcal{D} \subseteq [0,D-2]$ such that for all $h \geq h_0(A)$.

Theorems & Definitions (14)

  • Theorem 1: Nathanson nath1972-7nath96bb
  • Theorem 2
  • Theorem 3: Lev lev96b
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • Theorem 7
  • ...and 4 more