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Arithmetic Progressions in Sumsets and Lp-Almost-Periodicity

Published online by Cambridge University Press:  19 March 2013

ERNIE CROOT ,
IZABELLA ŁABA  and
OLOF SISASK
ERNIE CROOT
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA (e-mail: ecroot@math.gatech.edu)
IZABELLA ŁABA
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada (e-mail: ilaba@math.ubc.ca)
OLOF SISASK
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK (e-mail: O.Sisask@qmul.ac.uk)
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Abstract

We prove results about the Lp-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in Lp, and gives a very short proof of a theorem of Green that if A and B are subsets of {1,. . .,N} of sizes αN and βN then A+B contains an arithmetic progression of length at least

\begin{equation} \exp ( c (\alpha \beta \log N)^{1/2} - \log\log N). \end{equation}
Another almost-periodicity result improves this bound for densities decreasing with N: we show that under the above hypotheses the sumset A+B contains an arithmetic progression of length at least
\begin{equation} \exp\biggl( c \biggl(\frac{\alpha \log N}{\log^3 2\beta^{-1}} \biggr)^{1/2} - \log( \beta^{-1} \log N) \biggr). \end{equation}

Keywords

Primary 11B30Secondary 11B25
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 3 , May 2013 , pp. 351 - 365
Copyright
Copyright © Cambridge University Press 2013

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References

[1]Bourgain, J. (1990) On arithmetic progressions in sums of sets of integers. In A Tribute to Paul Erdős, Cambridge University Press, pp. 105–109.Google Scholar
[2]Chang, M.-C. (2002) A polynomial bound in Freiman's theorem. Duke Math. J. 113 399–419.Google Scholar
[3]Croot, E. and Sisask, O. (2010) A probabilistic technique for finding almost-periods of convolutions. Geom. Funct. Anal. 20 1367–1396.Google Scholar
[4]Green, B. (2002) Arithmetic progressions in sumsets. Geom. Funct. Anal. 12 584–597.Google Scholar
[5]Green, B. (2002) Restriction and Kakeya Phenomena, lecture notes. http://www.dpmms.cam.ac.uk/~bjg23/rkp.htmlGoogle Scholar
[6]Green, B. (2005) Finite field models in additive combinatorics. In Surveys in Combinatorics 2005, Vol. 327 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 1–27.Google Scholar
[7]Gut, A. (2005) Probability: A Graduate Course, Springer.Google Scholar
[8]Khintchine, A. (1923) Über dyadische Brüche. Math. Z. 18 109–116.Google Scholar
[9]Petridis, G. (2013) New proofs of Plünnecke-type estimates for product sets in groups. Combinatorica, doi: 10.1007/s00493-012-2818-5.Google Scholar
[10]Pisier, G. (1981) Remarques sur un résultat non publié de B. Maurey. In Seminar on Functional Analysis, 1980–1981, exp. no. 5, École Polytechnique, Palaiseau. http://www.numdam.org/item?id=SAF_1980-1981____A5_0Google Scholar
[11]Ruzsa, I. Z. (2009) Sumsets and structure. In Combinatorial Number Theory and Additive Group Theory, Springer, pp. 87–210.Google Scholar
[12]Sanders, T. (2008) Additive structures in sumsets. Math. Proc. Cambridge Philos. Soc. 144 289–316.Google Scholar
[13]Sanders, T. (2011) Green's sumset problem at density one half. Acta Arith. 146 91–101.Google Scholar
[14]Sanders, T. (2012) On the Bogolyubov–Ruzsa lemma. Anal. PDE 5 627–655.Google Scholar
[15]Tao, T. and Vu, V. H. (2006) Additive Combinatorics, Cambridge University Press.Google Scholar