• DOI:10.1007/BF02789444
• Corpus ID: 119664036

The Gaussian primes contain arbitrarily shaped constellations

@article{Tao2005TheGP,
  title={The Gaussian primes contain arbitrarily shaped constellations},
  author={Terence Tao},
  journal={Journal d’Analyse Math{\'e}matique},
  year={2005},
  volume={99},
  pages={109-176}
}

• T. Tao
• Published 20 January 2005
• Mathematics
• Journal d’Analyse Mathématique

We show that the Gaussian primesP[i] ⊆ ℤ[i] contain infinitely constellations of any prescribed shape and orientation. More precisely, we show that given any distinct Gaussian integersv0,…,vk−1, there are infinitely many sets {a+rv0,…,rvk−1}, witha ∈ℤ[i] andr ∈ℤ{0}, all of whose elements are Gaussian primes.The proof is modeled on that in [9] and requires three ingredients. The first is a hypergraph removal lemma of Gowers and Rödl-Skokan or, more precisely, a slight strenghthening of this lemma which can be found in [22]; this hypergraph removal lemma can be thought of as a generalization of the Szemerédi-Furstenberg-Katznelson theorem concerning multidimensional arithmetic progressions. The second ingredient is the transference argument from [9], 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, similar to that in [9], which yields a pseudorandom measure. which is concentrated on Gaussian “almost primes”.