On the Quantitative Bounds of Nonlinear Roth Theorem

We prove that any subset $A\subseteq \{1,\dots ,N\}$ lacking non-trivial polynomial progressions $(x,x+y,x+y^2)$ satisfies $\mid A\mid \le CN\exp (-c(\log \log N{)}^{\beta })$ 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{é}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{ő}s–Tur{á}n–Koksma equidistribution.