Abstract
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 integersv 0,…,v k−1, there are infinitely many sets {a+rv 0,…,rv k−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”.
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Tao, T. The Gaussian primes contain arbitrarily shaped constellations. J. Anal. Math. 99, 109–176 (2006). https://doi.org/10.1007/BF02789444
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DOI: https://doi.org/10.1007/BF02789444