On the Quantitative Bounds of Nonlinear Roth Theorem

Abstract

We prove that any subset $A \subseteq \{1, \dots, N\}$ lacking non-trivial polynomial progressions $(x, x + y, x + y^2)$ satisfies $\lvert A \rvert \le CN \exp\bigl(-c(\log \log N)^\beta\bigr)$ for effective absolute constants $C, c, \beta>0$. The argument develops a density increment framework localized to anisotropic quadratic Bohr sets. To handle mixed linear and quadratic frequencies, we establish a multilinear transference principle coupled with a relative quadratic inverse theorem based on Balog--Szemer{\'e}di--Gowers additive energy rigidity. Boundary stability across increments is maintained via semi-algebraic model regularization using cylindrical algebraic decomposition and Cauchy--Crofton estimates, while dimension growth is controlled by dissociated basis reduction and multidimensional Erd{\H{o}}s--Tur{\'a}n--Koksma equidistribution.