Quantitative bounds in the non-linear Roth theorem

We show that there exists c>0 such that any subset of {1,…,N} of density at least (loglogN)^{-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.