Florian Richter, Donald Robertson and I have uploaded to the arXiv our paper entitled A proof of the Erdős sumset conjecture. The main goal of the paper is to prove the following theorem, which verifies a conjecture of Erdős discussed in this previous post.

Theorem 1Let have positive upper density. Then it contains a sumset for infinite sets and .

As usual, the upper density of a set is the quantity

More generally, given a sequence of intervals of integers with lengths tending to infinity (or even more generally, given any Følner sequence in ), we define the upper density with respect to as

When the limit exists we say that the density of with respect to exists and denote it by (and whenever we write we are implicitly stating that the limit exists).

As explained in my earlier post mentioned above (see Proposition 12), one can adapt an argument of Di Nasso, Goldbring, Jin, Leth, Lupini and Mahlburg to reduce Theorem 1 to the following statement.

Theorem 2If has positive upper density, then there exist a sequence of intervals, a set and such that for every finite subset , the intersection

In my previous post I described how Theorem 2 can then be easily checked for weak mixing sets and for Bohr sets, whose definitions we now recall.

Definition 3A set is:

- a
Bohr setif there exist , a vector and an open set such that .- a weak mixing set with respect to the sequence of intervals if the set has density .

It is well known in the realm of ergodic theory that weak mixing and almost periodicity are extreme points of a dichotomy (see for instance, this post) and so it was natural to try to patch these two observations into a proof of Theorem 2 in general. While this is ultimately how we prove Theorem 2, there were several problems with implementing this idea. Below the fold I explain the problems that arise and (briefly) sketch how we address them. Continue reading