English

Infinite Sumsets in Sets with Positive Density

Dynamical Systems 2024-02-23 v2 Combinatorics Number Theory

Abstract

Motivated by questions asked by Erdos, we prove that any set ANA\subset{\mathbb N} with positive upper density contains, for any kNk\in{\mathbb N}, a sumset B1++BkB_1+\cdots+B_k, where B1,,BkNB_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=2k=2.

Keywords

Cite

@article{arxiv.2206.01786,
  title  = {Infinite Sumsets in Sets with Positive Density},
  author = {Bryna Kra and Joel Moreira and Florian K. Richter and Donald Robertson},
  journal= {arXiv preprint arXiv:2206.01786},
  year   = {2024}
}

Comments

49 Pages

R2 v1 2026-06-24T11:38:49.354Z