Quantitative bounds in the nonlinear Roth theorem

We show that there exists c > 0 such that any subset of { 1 , … , 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.