Quasipolynomial Bounds for Arithmetic Progressions

We prove Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. More quantitatively, for every fixed $k\ge3$, we show \[ r_k(N) C_kN(-c_k( N)^{_k}) \] with $C_k,c_k,\varepsilon_k>0$, where $r_k(N)$ is the largest size of a subset of $\{1,\ldots,N\}$ with no nonconstant $k$-term arithmetic progression.