Arithmetic progressions in sums of subsets of sparse sets
Arithmetic progressions in sums of subsets of sparse sets
复制标题
稀疏集子集之和的算术级数
DOI：
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发表时间：
2011
期刊：
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通讯作者：
T. Schoen
中科院分区：
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作者：
T. Schoen
A的子集和集，L（S）表示S中最长等差数列的长度。在S（A）（或一般在和集）中寻找大的算术结构的问题是组合数论中最基本的问题之一。它已经被深入研究，特别是在足够稠密的集合A ∈ {1，. . .，n}（|一|[1][2][3][4][5][6][7][8][9][10]最近Szemerédi和Vu在[6]，[7]和[8]中给出了多项式大小集合的这个问题的完整解决方案。除其他外，他们证明，如果A [n]和|一|d n1/d，其中d ≥ 2是固定整数，则
the subsets sumset of A and let L(S) stand for the length of the longest arithmetic progression in S. The problem of finding large arithmetic structures in S(A) (or generally in sumsets) is one of the most fundamental in combinatorial number theory. It has been intensively studied, especially in the case of sufficiently dense sets A ⊆ {1, . . . , n} (|A| ≥ nα for some α > 1/2; see for example [1], [3], [4], [5]). A complete solution of this problem for sets of polynomial size was given recently by Szemerédi and Vu in [6], [7] and [8]. They proved, among other things, that if A ⊆ [n] and |A| d n1/d, where d ≥ 2 is a fixed integer, then