The Gaussian primes contain arbitrarily shaped constellations

Abstract

We show that the Gaussian primes $P[i] \subseteq \Z[i]$ contain infinitely constellations of any prescribed shape and orientation. More precisely, given any distinct Gaussian integers $v_0,...,v_{k-1}$, we show that there are infinitely many sets $\{a+rv_0,...,a+rv_{k-1}\}$, with $a \in \Z[i]$ and $r \in \Z \backslash \{0\}$, all of whose elements are Gaussian primes. The proof is modeled on a recent paper by Green and Tao and requires three ingredients. The first is a hypergraph removal lemma of Gowers and R\"odl-Skokan; this hypergraph removal lemma can be thought of as a generalization of the Szemer\'edi-Furstenberg-Katznelson theorem concerning multidimensional arithmetic progressions. The second ingredient is the transference argument of Green and Tao, which allows one to extend this hypergraph removal lemma to a relative version, weighted by a pseudorandom measure. The third ingredient is a Goldston-Yildirim type analysis for the Gaussian integers, which yields a pseudorandom measure which is concentrated on Gaussian "almost primes".