Quantitative bounds for the nonlinear Roth configuration with a three-term progression

Establish quantitative upper bounds for the maximal size of a subset of the integers that contains no nontrivial configuration of the form x, x+y, x+2y, x+y^2, thereby resolving the configuration that remains beyond the reach of the methods discussed for higher-complexity polynomial Szemerédi problems.

Background

The paper studies quantitative bounds in the polynomial Szemerédi theorem and proves such a bound for the configuration x, x+y, x+2y, x+y3. The authors contrast this result with the closely related configuration x, x+y, x+2y, x+y2, which combines a three-term arithmetic progression with the nonlinear Roth configuration and has higher algebraic complexity.

The cited obstruction is that the relevant counting operator requires stronger inverse-theorem control, involving U3-norms for the first three functions and a U2-norm for the fourth. The transference method is not applicable to this pattern, while the previously used degree-lowering approaches do not provide the required estimates. Consequently, the paper treats the cubic configuration as a testing ground rather than resolving the quadratic configuration.

References

One of the easiest examples of a nontransferable pattern is $x,x+y,x+2y,x+y2$, which is a simultaneous consideration of the three term arithmetic progression and the nonlinear Roth configuration $x,x+y,x+y2$. This configuration however remains out of reach, as each of its terms has complexity at least $1$, hence the best inverse theorems for the counting operator

— Further bounds in the polynomial Szemerédi theorem  (2610.12032 - Bychawski et al., 8 Oct 2026) in Introduction