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Arithmetic progressions in sparse sumsets. (English) Zbl 1178.11011

Landman, Bruce (ed.) et al., Combinatorial number theory. Proceedings of the 2nd integers conference 2005 in celebration of the 70th birthday of Ronald Graham, Carrollton, GA, USA, October 27–30, 2005. Berlin: Walter de Gruyter (ISBN 978-3-11-019029-8/hbk). 157-164 (2007) and Integers 7, No. 2, Paper A10 (2007).
Let \(A\not=\emptyset\) be a finite set of integers. By Freiman’s theorem, if \(| 2A| \leq c| A| \) (where \(2A=A+A=\{a+a':\,a,a'\in A\}\)) then \(A\) is a large subset of a multidimensional arithmetic progression. In the paper under review, the authors show that if \(| A-A| =C| A| \) and \(| A-2A| =K| A| \) then \[ \frac{L(A-A)-1}2\geq\frac{\log| A| }{\log K} \] and \[ \frac{L(2A)-1}2\geq\frac{\log(| A| /| C| )}{\log \min\{CK,C^4\}}, \] where \(L(B)\) denotes the length of the longest arithmetic progression in \(B\). Consequently, if \(k\geq3\) is odd and \(N\) is sufficiently large then \(L(A+B)\geq k\) for any \(A,B\subseteq\{1,\ldots,N\}\) with \(| A| \cdot| B| \geq 6N^{2-2/(k-1)}\). This yields a new contribution not covered by B. Green’s result [Geom. Funct. Anal., 12, No. 3, 584–597 (2002; Zbl 1020.11009)]. The authors also give a construction of sets \(A\) such that \(2A\) contains no long arithmetic progression.
In contrast with previous methods involving harmonic analysis, the method of the present paper is completely elementary and hence the paper is more readable.
For the entire collection see [Zbl 1108.00016].

MSC:

11B25 Arithmetic progressions
11P70 Inverse problems of additive number theory, including sumsets

Citations:

Zbl 1020.11009
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