A Szemerédi-type regularity lemma in abelian groups, with applications
A Szemerédi-type regularity lemma in abelian groups, with applications
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阿贝尔群中的 Szemerédi 型正则引理及其应用
DOI：
10.1007/s00039-005-0509-8
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发表时间：
2003
期刊：
影响因子：
--
通讯作者：
B. Green
中科院分区：
文献类型：
--
作者：
B. Green
摘要。Szemerédi的规律性引理是图理论中的重要工具，在本文中，我们证明了Szemerédi的规律性引理在Abelian群体中的类似物，并在添加数字中得出了一些结果结构定理的集合几乎不含
$$ a \ subseteq \ {1，\ ldots，n \} $$具有δn2 triples（a1，a2，a3），其中a1 + a2 = a3 t，然后a = b c c，其中b是bus ne um-free and | | c | =δ'n，和
$$ \ delta^{\ prime} \ rightarrow 0 $$ as
$$ \ delta \ rightarrow 0
$$ \ alpha，\ epsilon> 0，$$如果
$$ n \，> \，n_ {0}（\ alpha，\ epsilon）$$，如果
$$ a \ subseteq \ {1，\ ldots，n \} $$具有大小αn，然后有一些d≠0，以至少包含
$$（\ alpha^{3} - \ epsilon）n $$三项算术进度，具有共同的差异d。
Abstract.Szemerédi’s regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemerédi’s regularity lemma in the context of abelian groups and use it to derive some results in additive number theory. One is a structure theorem for sets which are almost sum-free. If
$$A \subseteq \{1,\ldots,N\}$$ has δ N2 triples (a1, a2, a3) for which a1 + a2 = a3 then A = B ∪ C, where B is sum-free and |C| = δ′N, and
$$\delta^{\prime} \rightarrow 0$$ as
$$\delta \rightarrow 0.$$ Another answers a question of Bergelson, Host and Kra. If
$$\alpha, \epsilon > 0,$$ if
$$N\,>\,N_{0}(\alpha, \epsilon)$$ and if
$$A \subseteq \{1,\ldots,N\}$$ has size α N, then there is some d ≠ 0 such that A contains at least
$$(\alpha^{3}-\epsilon)N$$ three-term arithmetic progressions with common difference d.