Mathematics – Combinatorics
Scientific paper
2005-01-20
J. d.Analyse Mathematique 99 (2006), 109-176
Mathematics
Combinatorics
58 pages, no figures. An issue (pointed out by Lilian Matthiesen) regarding the need to ensure the linear forms are not commen
Scientific paper
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".
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