×

Additive structures in sumsets. (English) Zbl 1241.11010

Let \(A,A'\subset {\mathbb Z}/N{\mathbb Z}\) and \(A+A'=\{a+a': a\in A, a'\in A'\}\). The following theorems form the main results of the paper:
(1) Suppose that \(A_1,A_2\subset {\mathbb Z}/N{\mathbb Z}\) and that the geometric mean \(\alpha\) of the densities of the \(A_i\)’s is positive. Then \(A_1+A_2\) contains an arithmetic progression of length at least \(\exp(((\alpha^2\log N)^{(1/2)}-\log \alpha^{-1}\log\log N))\) for some absolute constant \(c>0\).
(2) Suppose that \(m\geq 3\) and \(A_1,\dots, A_m\subset {\mathbb Z}/N{\mathbb Z}\) and that \(\alpha\) denotes the geometric mean of the densities of the \(A_i\)’s. Then \(A_1+\dots+A_m\) contains an arithmetic progression of length \(c\alpha^{Cm^3\alpha^{1/(m-2)}N^{cm^{-2}\alpha^{1/(m-2)}}}\) for some absolute constants \(C,c>0\).
The second result strengthens the previously known explicitly or implicitly published results by B. Green [Geom. Funct. Anal. 12, No. 3, 584–597 (2002; Zbl 1020.11009)] or by M.-Ch. Chang [Duke Math. J. 113, No. 3, 399–419 (2002; Zbl 1035.11048)] for \(m=3\) and \(A_1=A_2=A_3\), or \(m=4\) and \(A_1=A_2=A\), \(A_3=A_4=-A\), respectively. The case \(m=2\) is much harder in comparison with the case \(m\geq 3\). So the first result is weaker than that proved previously by Green [loc. cit.], but as the author believes his new proof method of (1) is conceptually simpler though technically more challenging.

MSC:

11B25 Arithmetic progressions
11B13 Additive bases, including sumsets
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] DOI: 10.1215/S0012-7094-02-11331-3 · Zbl 1035.11048 · doi:10.1215/S0012-7094-02-11331-3
[2] DOI: 10.1007/s00039-002-8258-4 · Zbl 1020.11009 · doi:10.1007/s00039-002-8258-4
[3] DOI: 10.1007/BF01876039 · Zbl 0816.11008 · doi:10.1007/BF01876039
[4] DOI: 10.1007/s000390050105 · Zbl 0959.11004 · doi:10.1007/s000390050105
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.