Infinite Sumsets in Sets with Positive Density

TL;DR

This paper proves that sets of natural numbers with positive density contain complex sumsets formed by infinite subsets, using novel ergodic theory techniques, extending previous results to arbitrary sums.

Contribution

It introduces new ergodic theory methods to show that positive density sets contain sumsets of any finite number of infinite subsets, advancing understanding of additive combinatorics.

Findings

Positive density sets contain sumsets of any finite number of infinite subsets.

New ergodic theory techniques are developed for this proof.

Results extend previous work from the case of two sumsets to arbitrary finite sums.

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 $k=2$.