Long arithmetic progressions in sumsets: Thresholds and bounds
Long arithmetic progressions in sumsets: Thresholds and bounds
复制标题
求和集中的长算术级数：阈值和界限
DOI：
10.1090/s0894-0347-05-00502-3
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发表时间：
2005
影响因子：
3.9
通讯作者：
V. Vu
中科院分区：
文献类型：
--
作者：
E. Szemerédi;V. Vu
L∗A={A1+···+al|ai∈Ai，ai=aj}。在所有数学中最著名的结果之一是维诺格拉多夫定理，它说3P(P是素数集)包含所有足够大的奇数，以及瓦林猜想(由希尔伯特、哈代和利特尔伍德、华和许多其他人证明)，它断言对于任何给定的r，存在一个数L，使得L∗Nr(N表示第r次方次方的集合)包含所有足够大的正整数(参见[29]，以获得关于这些结果的精彩论述)。近年来，有限和集的研究受到了相当大的关注。给定一个有限集A和一个正整数L，维诺加多夫-瓦林结果的自然类比是：在适当的条件下，和集A(L∗A)包含一个长的算术级数。设A是区间[n]={1，.。。，n}，其中n是大的正整数。我们要解决的具体问题是估计作为L、n和|A|的函数的La(L∗A)中最长算术级数的最小长度。我们用f(|A|，L，n)(f∗(|A|，L，n))表示这个函数，遵循[13]中的记号。Bourain，Freiman，Halberstam，Green，Ruzsa和Sarkozy已经发现了f(|A|，L，n)的许多估计(见第三节)，但这些结果大多集中在密度很高的集合上，即|A|接近n，估计f∗(|A|，L，n)似乎困难得多，在我们研究之前知之甚少。
l∗A = {a1 + · · ·+ al|ai ∈ Ai, ai = aj}. Among the most well-known results in all of mathematics are Vinogradov’s theorem, which says that 3P (P is the set of primes) contains all sufficiently large odd numbers, and Waring’s conjecture (proved by Hilbert, Hardy and Littlewood, Hua, and many others), which asserts that for any given r, there is a number l such that l∗Nr (N denotes the set of rth powers) contains all sufficiently large positive integers (see [29] for an excellent exposition concerning these results). In recent years, a considerable amount of attention has been paid to the study of finite sumsets. Given a finite set A and a positive integer l, the natural analogue of Vinogadov-Waring results is to show that under proper conditions, the sumset lA (l∗A) contains a long arithmetic progression. Let us assume that A is a subset of the interval [n] = {1, . . . , n}, where n is a large positive integer. The concrete problem we would like to address is to estimate the minimum length of the longest arithmetic progression in lA (l∗A) as a function of l, n, and |A|. We denote this function by f(|A|, l, n) (f∗(|A|, l, n)), following the notation in [13]. Many estimates for f(|A|, l, n) have been discovered by Bourgain, Freiman, Halberstam, Green, Ruzsa, and Sarkozy (see Section 3), but most of these results focus on sets with very high density, namely |A| is close to n. Estimating f∗(|A|, l, n) seems much harder, and not much was known prior to our study.