Additive structure via random sampling

Questions about set addition have been studied for centuries, particularly in number theory.

There are for instance many famous conjectures about addition and the primes, such as the Goldbach conjecture and the twin prime conjecture, but there are also many theorems, such as Vinogradov´s result that any sufficiently large odd integer can be written as a sum of three primes, and van der Corput´s result that the primes contain infinitely many non-trivial three-term arithmetic progressions (3APs).

These results were both from the 1930s, and around that time, people started asking about the addition of quite general sets of integers, as opposed to specific ones like the primes.

Following foundational results of Roth, it was one of Erdos´s favourite conjectures that if A is any set of integers that has about the same number of elements among 1,2,...,N as the primes (about N/log N), then A must contain infinitely many 3APs, and, in fact, arithmetic progressions of any length.The 3AP-case of this conjecture was recently proved by Bloom and the applicant, and very recently a sensational strengthening was proved by Kelley and Meka, with some subsequent extensions by Bloom and the applicant.

This proposal aims to harness these advances to make progress both on Roth-type problems as well as other problems, to strengthen the random sampling-based methods from both papers by attempting to combine their underlying building blocks, and to explore the limitations of these tools.