Improved Bounds for Szemerédi's Theorem

Let $r_k(N)$ denote the size of the largest subset of $[N]=\{1,\dots ,N\}$ with no $k$-term arithmetic progression. We show that for $k\ge 5$, there exists $c_k>0$ such that $$r_k(N)\ll N\exp (-(\log \log N{)}^{c_k})\mathrm{.}$$ Our proof is a consequence of recent quasipolynomial bounds on the inverse theorem for the Gowers $U^k$-norm as well as the density increment strategy of Heath-Brown and Szemerédi as reformulated by Green and Tao.