Bounds in a popular multidimensional nonlinear Roth theorem

TL;DR

This paper extends Roth's theorem to multidimensional settings with nonlinear configurations, providing effective bounds and a popular version showing many such configurations exist in dense sets.

Contribution

It introduces a multidimensional nonlinear Roth theorem and an effective popular version with near-optimal configuration counts, advancing understanding of polynomial configurations.

Findings

Established a multidimensional nonlinear Roth theorem.

Proved an effective popular version with exponential bounds.

Showed dense sets contain many nonlinear configurations.

Abstract

A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form $x$, $x+d$, $x+d^2$. We obtain a multidimensional version of this result, which can be regarded as a first step towards effectivising those cases of the multidimensional polynomial Szemer\'edi theorem involving polynomials with distinct degrees. In addition, we prove an effective ``popular'' version of this result, showing that every dense set has some non-zero $d$ such that the number of configurations with difference parameter $d$ is almost optimal. Perhaps surprisingly, the quantitative dependence in this result is exponential, compared to the tower-type bounds encountered in the popular linear Roth theorem.