An Arithmetic Regularity Lemma, An Associated Counting Lemma, and Applications
An Arithmetic Regularity Lemma, An Associated Counting Lemma, and Applications
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算术正则引理、相关计数引理及应用
DOI：
10.1007/978-3-642-14444-8_7
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
T. Tao
中科院分区：
文献类型：
--
作者：
B. Green;T. Tao
在他的70岁生日时，Szemeredi的规律性引理可以看作是任意密集图的粗糙结构定理，将这样的图形分解为结构化的零件，我们建立了一个均匀的规律性，将这些图形分解为一个粗糙的结构。类似地分解有界函数f：[n]→ℂ，进入（良好的分配，虚拟）s-step nil sequence，l2中很小的误差，进一步在gowers+1-norm中是微小的误差，其中s≥1是一个参数。
To Endre Szemeredi on the occasion of his 70th birthday Szemeredi’s regularity lemma can be viewed as a rough structure theorem for arbitrary dense graphs, decomposing such graphs into a structured piece, a small error, and a uniform piece. We establish an arithmetic regularity lemma that similarly decomposes bounded functions f: [N] →ℂ, into a (well-equidistributed, virtual) s-step nilsequence, an error which is small in L2 and a further error which is minuscule in the Gowers Us+1-norm, where s ≥ 1 is a parameter. We then establish a complementary arithmetic counting lemma that counts arithmetic patterns in the nilsequence component of f.