New lower bounds for $r_3(N)$

Abstract

We develop recent ideas of Elsholtz, Proske, and Sauermann to construct denser subsets of $\{1,\dots ,N\}$ that lack arithmetic progressions of length $3$. This gives the first quasipolynomial improvement since the original construction of Behrend.

11 pages, comments welcome! (in v2 we have corrected some typos in our proof of Proposition 1.3) v3 update: We have now merged our results with Elsholtz, Proske, and Sauermann, see arXiv:2401.12802