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- W2965893648 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 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 Rodl-Skokan; this hypergraph removal lemma can be thought of as a generalization of the Szemeredi-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." @default.
- W2965893648 created "2019-08-13" @default.
- W2965893648 creator A5066728260 @default.
- W2965893648 date "2005-01-20" @default.
- W2965893648 modified "2023-09-27" @default.
- W2965893648 title "Information theory, relative versions of the hypergraph regularity and removal lemmas, the Szemer'edi-Furstenberg-Katznelson theorem, and prime constellations in number fields" @default.
- W2965893648 hasPublicationYear "2005" @default.
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