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Problems in additive number theory, VI: Sizes of sumsets

Melvyn B. Nathanson

TL;DR

The paper investigates the range of sizes of $h$-fold sumsets and restricted sumsets for finite subsets of the integers and more general additive abelian groups. It establishes sharp universal bounds, analyzes how these ranges behave in ordered groups and groups of unbounded exponent, and computes complete descriptions in key cases (notably $\\mathcal{R}_{\\mathbb{Z}}(h,3)$). It reveals that sumset-size sets can fail to be intervals for $h\\ge3$, and develops a framework of sumset-size trajectories $\\kappa_{\infty}(A)$ and related sets to study growth patterns. The results provide both exact formulas and structural insights, with numerous open questions on higher-$h$ and constrained trajectories that guide future work in additive combinatorics.

Abstract

This paper describes problems concerning the range of cardinalities of sumsets and restricted sumsets of finite subsets of the integers and finite subsets of ordered abelian groups.

Problems in additive number theory, VI: Sizes of sumsets

TL;DR

The paper investigates the range of sizes of -fold sumsets and restricted sumsets for finite subsets of the integers and more general additive abelian groups. It establishes sharp universal bounds, analyzes how these ranges behave in ordered groups and groups of unbounded exponent, and computes complete descriptions in key cases (notably ). It reveals that sumset-size sets can fail to be intervals for , and develops a framework of sumset-size trajectories and related sets to study growth patterns. The results provide both exact formulas and structural insights, with numerous open questions on higher- and constrained trajectories that guide future work in additive combinatorics.

Abstract

This paper describes problems concerning the range of cardinalities of sumsets and restricted sumsets of finite subsets of the integers and finite subsets of ordered abelian groups.

Paper Structure

This paper contains 8 sections, 10 theorems, 186 equations.

Table of Contents

  1. A remark of Erdős and Szemerédi
  2. Upper and lower bounds for sumset sizes in groups
  3. Ordered groups and groups of unbounded exponent
  4. Sizes of 2-fold sumsets and 2-fold restricted sumsets
  5. Missing sumset sizes
  6. The sumset size set $\mathcal{R}_{\mathbf Z}(h,3)$
  7. Sequences of sumset sizes
  8. On $|hA| = |A|$

Key Result

Theorem 1

Let $G$ be an additive abelian group. For all integers $h\geq 2$ and $k\geq 2$, there are best possible upper bounds and and best possible lower bounds and Moreover, for all $k \geq 1$, and, for all $h \geq 1$,

Theorems & Definitions (20)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 10 more