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Plünnecke inequalities for measure graphs with applications

Published online by Cambridge University Press:  06 October 2015

KAMIL BULINSKI Open the ORCID record for KAMIL BULINSKI [Opens in a new window]  and
ALEXANDER FISH
KAMIL BULINSKI
Affiliation:
School of Mathematics and Statistics, University of Sydney, Australia email kbul7081@uni.sydney.edu.au, alexander.fish@sydney.edu.au
ALEXANDER FISH
Affiliation:
School of Mathematics and Statistics, University of Sydney, Australia email kbul7081@uni.sydney.edu.au, alexander.fish@sydney.edu.au
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Abstract

We generalize Petridis’s new proof of Plünnecke’s graph inequality to graphs whose vertex set is a measure space. Consequently, by a recent work of Björklund and Fish, this gives new Plünnecke inequalities for measure-preserving actions which enable us to deduce, via a Furstenberg correspondence principle, Banach density estimates in countable abelian groups that extend those given by Jin.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 2 , April 2017 , pp. 418 - 439
Copyright
© Cambridge University Press, 2015 

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References

Beiglböck, M., Bergelson, V. and Fish, A.. Sumset phenomenon in countable amenable groups. Adv. Math. 223(2) (2010), 416–432.CrossRefGoogle Scholar
Björklund, M. and Fish, A.. Plünnecke inequalities for countable abelian groups. J. Reine Angew. Math. to appear. Preprint, 2013, arXiv:1311.5372v2.Google Scholar
Björklund, M. and Fish, A.. Product set phenomena for countable groups. Adv. Math. 275 (2015), 47–113.CrossRefGoogle Scholar
Furstenberg, H.. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Anal. Math. 31 (1977), 204–256.CrossRefGoogle Scholar
Jin, R.. Plünnecke’s theorem for asymptotic densities. Trans. Amer. Math. Soc. 363(10) (2011), 5059–5070.CrossRefGoogle Scholar
Petridis, G.. Plünnecke’s inequality. Combin. Probab. Comput. 20(6) (2011), 921–938.CrossRefGoogle Scholar
Plünnecke, H.. Eine zahlentheoretische Anwendung der Graphentheorie. J. Reine Angew. Math. 243 (1970), 171–183.Google Scholar
Ruzsa, I. Z.. Sumsets and structure. Combinatorial Number Theory and Additive Group Theory (Advanced Courses in Mathematics CRM Barcelona) . Birkhäuser, Basel, 2009, pp. 87–210.Google Scholar
Tao, T. and Vu, V. H.. Additive Combinatorics (Cambridge Studies in Advanced Mathematics, 105) . Cambridge University Press, Cambridge, 2010, paperback edition.Google Scholar