Quantitative bounds in the nonlinear Roth theorem

Peluse, S. and Prendiville, S. (2024) Quantitative bounds in the nonlinear Roth theorem. Inventiones Mathematicae, 238 (3). pp. 865-903. ISSN 0020-9910

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Abstract

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.