The Gaussian primes contain arbitrarily shaped constellations
The Gaussian primes contain arbitrarily shaped constellations
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高斯素数包含任意形状的星座
DOI：
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发表时间：
2005
期刊：
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通讯作者：
T. Tao
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文献类型：
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作者：
T. Tao
我们证明了高斯素数P [i] ≠ P [i]包含任意给定形状和方向的无穷星座。更精确地说，我们证明了给定任何不同的高斯整数sv 0，.，vk −1，存在无穷多个集合{a+ rv 0，.，rvk −1}，其中a ∈ <$[i]且r ∈ <${0}，其所有元素都是高斯素数。第一个是Gowers和Rödl-Skokan的超图移除引理，或者更确切地说，是这个引理的一个轻微的加强，可以在[22]中找到;这个超图移除引理可以被认为是关于多维算术级数的Szemerédi-Furstenberg-Katznelson定理的推广。第二个成分是来自[9]的转移参数，它允许将超图移除引理扩展到相对版本，由伪随机测度加权。第三个成分是高斯整数的Goldston-Yildirim类型分析，类似于[9]中的分析，它产生伪随机测度。这是集中在高斯“几乎素数”。
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”.