Bounds in a popular multidimensional nonlinear Roth theorem

A nonlinear version of Roth's theorem states that dense sets of integers contain configurations of the form x, $x+d$, x+d². 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.