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Extremal problems and the combinatorics of sumsets

Melvyn B. Nathanson

TL;DR

This survey synthesizes extremal problems in the combinatorics of sumsets, focusing on bases, nonbases, and Sidon-type sets. It surveys foundational and modern results on the growth and structure of sumsets, including the asymptotic forms of $hA$ for finite sets, chromatic and multiplicity refinements, and lattice-point analogues via Ehrhart theory. The work highlights central conjectures (notably Erd\H{o}s--Tur\'an) and advances in constructing thin and minimal bases, maximal nonbases, and perturbation-based Sidon configurations, as well as the role of approximate groups in understanding additive growth. It also extends the theory to linear forms and abelian semigroups, revealing deep connections between combinatorial additive number theory, discrete geometry, and algebraic growth phenomena, with numerous open problems and recent constructive breakthroughs.

Abstract

This is a survey of old and new problems and results in additive number theory.

Extremal problems and the combinatorics of sumsets

TL;DR

This survey synthesizes extremal problems in the combinatorics of sumsets, focusing on bases, nonbases, and Sidon-type sets. It surveys foundational and modern results on the growth and structure of sumsets, including the asymptotic forms of for finite sets, chromatic and multiplicity refinements, and lattice-point analogues via Ehrhart theory. The work highlights central conjectures (notably Erd\H{o}s--Tur\'an) and advances in constructing thin and minimal bases, maximal nonbases, and perturbation-based Sidon configurations, as well as the role of approximate groups in understanding additive growth. It also extends the theory to linear forms and abelian semigroups, revealing deep connections between combinatorial additive number theory, discrete geometry, and algebraic growth phenomena, with numerous open problems and recent constructive breakthroughs.

Abstract

This is a survey of old and new problems and results in additive number theory.
Paper Structure (10 sections, 22 theorems, 70 equations)

This paper contains 10 sections, 22 theorems, 70 equations.

Table of Contents

  1. Sumsets and asymptotic bases
  2. Thin bases
  3. Minimal asymptotic bases
  4. Maximal asymptotic nonbases
  5. Oscillations
  6. Sums of finite sets
  7. Lattice points and abelian semigroups
  8. Approximate groups
  9. Sidon sets and $B_h$-sets
  10. Sidon sets for linear forms

Key Result

Theorem 1

For $\alpha, \beta, \gamma \in \mathbf R$ with $\alpha > 0$ and $1/3 < \gamma \leq 1/2$, let for all $n \in \mathbf N$. For almost all $A \in \Omega$ and for all but finitely many pairs $(m,n)$ with $m \neq n$,

Theorems & Definitions (22)

  • Theorem 1: Erdős-Nathanson nath1986-54
  • Theorem 2: Nathanson nath1981-47
  • Theorem 3: Nathanson nath2022-201
  • Theorem 4: Erdős-Nathanson nath1975-13
  • Theorem 5: Erdős-Nathanson nath1979-38
  • Theorem 6: Erdős-Nathanson nath1988-63
  • Theorem 7: Nathanson nath2020-188
  • Theorem 8: Erdős-Nathanson nath1975-17
  • Theorem 9: Erdős-Nathanson nath1975-17
  • Theorem 10: Erdős-Nathanson nath1976-22
  • ...and 12 more