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Izvestiya: Mathematics, 2009, Volume 73, Issue 5, Pages 1033–1075
DOI: https://doi.org/10.1070/IM2009v073n05ABEH002472 (Mi im2657) 		 
This article is cited in 6 scientific papers (total in 6 papers)

On a two-dimensional analogue of Szemerédi's theorem in Abelian groups

I. D. Shkredov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
English version PDF (499 kB) Citations (6)
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DOI: https://doi.org/10.1070/IM2009v073n05ABEH002472
Abstract: Let $G$ be a finite Abelian group and $A\subseteq G\times G$ a set of cardinality at least $|G|^2/(\log\log|G|)^c$, where $c>0$ is an absolute constant. We prove that $A$ contains a triple $\{(k,m),(k+d,m),(k,m+d)\}$ with $d\neq0$. This is a two-dimensional generalization of Szemerédi's theorem on arithmetic progressions.
Keywords: two-dimensional generalizations of Szemerédi's theorem, problems on arithmetic progressions, Roth's theorem, Bohr sets.
Received: 03.05.2007
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2009, Volume 73, Issue 5, Pages 181–224
DOI: https://doi.org/10.4213/im2657
Bibliographic databases:
UDC: 511.34+511.218+511.336
MSC: 35J25, 37A15
Language: English
Original paper language: Russian
Citation: I. D. Shkredov, “On a two-dimensional analogue of Szemerédi's theorem in Abelian groups”, Izv. RAN. Ser. Mat., 73:5 (2009), 181–224; Izv. Math., 73:5 (2009), 1033–1075
Citation in format AMSBIB
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  • https://www.mathnet.ru/eng/im/v73/i5/p181
    • This publication is cited in the following 6 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    		Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    Abstract page:695
    Russian version PDF:199
    English version PDF:8
    References:49
    First page:16
     
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