A proof of Erd\H{o}s's $B+B+t$ conjecture

TL;DR

This paper proves that any positive density set of natural numbers can be shifted to contain a specific restricted sumset formed by an infinite subset, advancing understanding of sumset structures in dense sets.

Contribution

It establishes a new result linking positive density sets to the presence of restricted sumsets, specifically showing such sumsets can be embedded via shifts.

Findings

Positive density sets can be shifted to contain specific restricted sumsets.

Existence of infinite subsets B within A such that B+B minus the diagonal is contained in a shifted version of A.

Advances the understanding of sumset configurations in dense subsets of natural numbers.

Abstract

We show that every set $A$ of natural numbers with positive upper density can be shifted to contain the restricted sumset $\{b_1 + b_2 : b_1, b_2\in B \text{ and } b_1 \neq b_2 \}$ for some infinite set $B \subset A$.