We show that there exists $c>0$ such that any subset of $\{1, \dots , N\}$ of density at least $(\log \log {N})^{-c}$ contains a nontrivial progression of the form $x$ , $x+y$ , $x+y^{2}$ . This is the first quantitatively effective version of the Bergelson–Leibman polynomial Szemerédi theorem for a progression involving polynomials of differing degrees. Our key innovation is an inverse theorem characterising sets for which the number of configurations $x$ , $x+y$ , $x+y^{2}$ deviates substantially from the expected value. In proving this, we develop the first effective instance of a concatenation theorem of Tao and Ziegler, with polynomial bounds.
