Corners in dense subsets of P^d

Let ${\text{PP}}^d$ be the $d$-fold direct product of the set of primes. We prove that if $A$ is a subset of ${\text{PP}}^d$ of positive relative upper density then $A$ contains infinitely many “corners”, that is sets of the form $\{x,x+t{e}_1,\mathrm{.}\mathrm{.}\mathrm{.},x+t{e}_d\}$ where x is an integer point and e_1,…,e_d are the standard basis vectors of the d-dimensional Euclidean space. Our argument is based on proving a removal lemma for weighted uniform hypergraphs, where the weight system is defined in terms of a pairwise linearly independent family of linear forms.