The Gaussian primes contain arbitrarily shaped constellations

By Terence Tao. 2005

64 citations recorded by the source corpus.

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”.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c58635ad392e96995e9fbdc7f7a7a2bf42591d92

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Citations in the corpus

A relative Szemerédi theorem (2013, methodology): “We borrow most of our notation and definitions from Tao [45, 46].”
The dichotomy between structure and randomness, arithmetic progressions, and the primes (2005, background): “The argument in [59], which shows that the Gaussian primes (or any dense subset thereof) contains infinitely many constellations of any prescribed shape, and can be viewed as a twodimensional analogue of Theorem 1.”
The primes contain arbitrarily long polynomial progressions (2006, methodology): “ of polynomials which vanish at the origin. This is however still open even in the linear case, the key diﬃculty being that the tensor product of pseudorandom measures is not pseudorandom. In view of [36] however, it should be possible (though time-consuming) to obtain a counterpart to Theorem 1.3 f”