Improved Bounds for Szemerédi's Theorem

Let r_k(N) denote the size of the largest subset of [N] = {1,…,N} with no k-term arithmetic progression. We show that for k≥ 5, there exists c_k>0 such that r_k(N)≪ Nexp(-(loglog N)^c_k). 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.