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Izv. RAN. Ser. Mat., 2009, Volume 73, Issue 5, Pages 181–224 (Mi izv2657)  

This article is cited in 3 scientific papers (total in 3 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

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.

DOI: https://doi.org/10.4213/im2657

Full text: PDF file (771 kB)
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English version:
Izvestiya: Mathematics, 2009, 73:5, 1033–1075

Bibliographic databases:

UDC: 511.34+511.218+511.336
MSC: 35J25, 37A15
Received: 03.05.2007

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://doi.org/10.4213/im2657
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. I. D. Shkredov, “On the Gowers Norms of Certain Functions”, Math. Notes, 92:4 (2012), 554–569  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. Solymosi J., “Roth-Type Theorems in Finite Groups”, Eur. J. Comb., 34:8, SI (2013), 1454–1458  crossref  mathscinet  zmath  isi  elib
    3. Prendiville S., “Matrix Progressions in Multidimensional Sets of Integers”, Mathematika, 61:1 (2015), 14–48  crossref  mathscinet  zmath  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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