Factorization in generalized arithmetic progressions and application to the Erdos-Szemeredi sum-product problems
Factorization in generalized arithmetic progressions and application to the Erdos-Szemeredi sum-product problems
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DOI：
10.1007/s00039-003-0428-5
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发表时间：
2003-01-01
影响因子：
2.2
通讯作者：
Chang, M
中科院分区：
文献类型：
--
作者：
Chang, M
定理0。设A是一个足够大的有限整数集，使得|A+ A| ≤ C1|一|，（1.2）对于常数C1.则A包含在适当的d维算术数列P中，满足d≤[C1 - 1]，（1.3）|P| ≤ C2|一|其中C2= C2（C1）是取决于C1的常数。这一结果是近年来研究和改进的重点。参见例如[BL]、[Bi]、[Cl]、[CoZ]、[E1]、[ErS]、[F]、[FHR]、[G1]、[G2]、[H]、[KLT]、[KT]、[N]、[NT]、[R1，2，3]、[ST]和[T]。特别是在[C1]中表明，可以取（1.4）log C2%C21（log C1）3。(1.5)在[ErS]中，证明了和集A+ A或积集AA，A是整数的任意有限集，需要具有本质上极值的大小，在以下意义下max（|A+ A|,|AA|）> cε|一|2− ε对于所有ε> 0，（1.6）
Theorem 0. Let A be a sufficiently large finite set of integers such that| A+ A|≤ C1| A|,(1.2) for some constant C1. Then A is contained in a proper d-dimensional arithmetic progression P satisfying d≤[C1− 1],(1.3)| P|≤ C2| A|,(1.4) where C2= C2 (C1) is a constant depending on C1. This result has been the focus of research and improvements over recent years. See for instance [BL],[Bi],[C1],[CoZ],[E1],[ErS],[F],[FHR],[G1],[G2],[H],[KLT],[KT],[N],[NT],[R1, 2, 3],[ST], and [T]. It is shown in particular in [C1] that one may take in (1.4) log C2% C2 1 (log C1) 3.(1.5) In [ErS], it is conjectured that either the sumset A+ A or the product set AA, A being an arbitrary finite set of integers, needs to have essentially extremal size, in the following sense max (| A+ A|,| AA|)> cε| A| 2− ε for all ε> 0,(1.6)