  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 Szemer\'edi 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.
