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Infinite Sumsets in Sets with Positive Density

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

Abstract

Motivated by questions asked by Erdos, we prove that any set $A\subset{\mathbb N}$ with positive upper density contains, for any $k\in{\mathbb N}$, a sumset $B_1+\cdots+B_k$, where $B_1,\dots,B_k\subset{\mathbb N}$ are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of $k=2$.

Infinite Sumsets in Sets with Positive Density

Abstract

Motivated by questions asked by Erdos, we prove that any set with positive upper density contains, for any , a sumset , where are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of .
Paper Structure (28 sections, 51 theorems, 160 equations)

This paper contains 28 sections, 51 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topological systems.
  4. Measure preserving systems.
  5. Factor maps.
  6. Generic points and support of a measure.
  7. Conditional expectation.
  8. Disintegrations.
  9. Ergodicity and ergodic decompositions.
  10. Furstenberg's correspondence principle.
  11. A discussion of two summands
  12. Erdős cubes
  13. Defining Erdős cubes
  14. \ref{['thm:cubes_exist']} implies \ref{['thm:main_theorem']}
  15. Cube systems and pronilfactors
  16. ...and 13 more sections

Key Result

Theorem 1.1

If $A \subset \mathbb{N}$ has positive upper Banach density then for every $k \in \mathbb{N}$ there are infinite sets $B_1,\dots,B_k \subset \mathbb{N}$ with $B_1 + \cdots + B_k \subset A$.

Theorems & Definitions (121)

  • Theorem 1.1
  • Definition 1.2
  • Definition 2.1: Measurable factor map
  • Definition 2.2: Continuous factor map
  • Definition 2.3: Generic points
  • Lemma 2.4
  • proof
  • Theorem 2.5: Disintegrations over factor maps, see for example EW11
  • Corollary 2.6
  • Definition 2.7: Ergodic decomposition
  • ...and 111 more