Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Problems on infinite sumset configurations in the integers and beyond

Bryna Kra, Joel Moreira, Florian K. Richter, Donald Robertson

TL;DR

This survey investigates which infinite sumset configurations are guaranteed in sets of positive density and extends the inquiry to primes, random sets, and general amenable groups. It develops density analogs of Hindman-type results, refines sumset patterns (including $B+C$, $B_1+\cdots+B_k$, and ordered variants), and employs ergodic theory, uniformity norms, Følner sequences, and ultrafilters to map the landscape of possibilities and obstructions. Key contributions include the density-Hindman results of KMRR and related refinements (e.g., constraints on summands, density Ramsey properties, and recurrence-based constructions), along with a suite of obstacles such as Straus-type counterexamples and higher-order obstructions. The work also surveys breakthroughs in the primes (Green–Tao, Maynard–Zhang–Tao–Ziegler) and random-model contexts, and it outlines numerous open problems and conjectures that guide future research in infinitary density combinatorics and its extensions to general groups and ultrafilter frameworks.

Abstract

In contrast to finite arithmetic configurations, relatively little is known about which infinite patterns can be found in every set of natural numbers with positive density. Building on recent advances showing infinite sumsets can be found, we explore numerous open problems and obstructions to finding other infinite configurations in every set of natural numbers with positive density.

Problems on infinite sumset configurations in the integers and beyond

TL;DR

This survey investigates which infinite sumset configurations are guaranteed in sets of positive density and extends the inquiry to primes, random sets, and general amenable groups. It develops density analogs of Hindman-type results, refines sumset patterns (including , , and ordered variants), and employs ergodic theory, uniformity norms, Følner sequences, and ultrafilters to map the landscape of possibilities and obstructions. Key contributions include the density-Hindman results of KMRR and related refinements (e.g., constraints on summands, density Ramsey properties, and recurrence-based constructions), along with a suite of obstacles such as Straus-type counterexamples and higher-order obstructions. The work also surveys breakthroughs in the primes (Green–Tao, Maynard–Zhang–Tao–Ziegler) and random-model contexts, and it outlines numerous open problems and conjectures that guide future research in infinitary density combinatorics and its extensions to general groups and ultrafilter frameworks.

Abstract

In contrast to finite arithmetic configurations, relatively little is known about which infinite patterns can be found in every set of natural numbers with positive density. Building on recent advances showing infinite sumsets can be found, we explore numerous open problems and obstructions to finding other infinite configurations in every set of natural numbers with positive density.
Paper Structure (25 sections, 20 theorems, 126 equations, 1 figure)

This paper contains 25 sections, 20 theorems, 126 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Infinite sumsets in sets of integers with positive density
  3. Searching for a density version of Hindman's theorem
  4. Sums of distinct infinite sets
  5. Refinements of sumsets in positive density
  6. Constraining the summands
  7. Combinatorial obstructions to $B+B+t$
  8. Ordered sums
  9. Sumsets of the form B+mB
  10. Uniformity norms and shifts
  11. The density Ramsey property
  12. Sumsets and recurrence
  13. Sumsets in sets of integers without density
  14. Sumsets in the primes
  15. Zero density sets
  16. ...and 10 more sections

Key Result

Theorem 2.2

Every set $A\subset\mathbb{N}$ of positive upper Banach density contains a sumset for some shift $t\geq0$ and some infinite set $B \subset A$.

Figures (1)

  • Figure 1: Summary of the relations among the questions (assuming positive answers), conjectures, and theorems in \ref{['sec:sumsets_integers']}.

Theorems & Definitions (63)

  • Example 2.1: Straus example
  • Theorem 2.2: KMRR2
  • Example 2.3
  • Conjecture 2.4: KMRR2
  • Conjecture 2.5
  • Proposition 2.6: See also Hindman79
  • proof
  • Theorem 2.7: MRR
  • Theorem 2.8: KMRR
  • Conjecture 2.9
  • ...and 53 more