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Some Properties of Lower Level-Sets of Convolutions

Published online by Cambridge University Press:  12 April 2012

ERNIE CROOT
ERNIE CROOT*
Affiliation:
School of Mathematics, Georgia Institute of Technology, 103 Skiles, Atlanta, GA 30332, USA (e-mail: [email protected])
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Abstract

In the present paper we prove a certain lemma about the structure of ‘lower level-sets of convolutions’, which are sets of the form {x ∈ ℤN : 1A * 1A(x) ≤ γ N} or of the form {x ∈ ℤN : 1A(x) < γ N}, where A is a subset of ℤN. One result we prove using this lemma is that if |A| = θ N and |A+A| ≤ (1 − ϵ)N, 0 < ϵ < 1, then this level-set contains an arithmetic progression of length at least Nc, c = c(θ, ϵ, γ) > 0. It is perhaps possible to obtain such a result using Green's arithmetic regularity lemma (in combination with some ideas of Bourgain [6]); however, our method of proof allows us to obtain non-tower-type quantitative dependence between the constant c and the parameters θ and ϵ. For various reasons (discussed in the paper) one might think, wrongly, that such results would only be possible for level-sets involving triple and higher convolutions.

Keywords

Primary 11B30
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 4 , July 2012 , pp. 515 - 530
Copyright
Copyright © Cambridge University Press 2012

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References

[1]Ajtai, M., Iwaniec, H., Komlós, J., Pintz, J. and Szemerédi, E. (1990) Construction of a thin set with small Fourier coefficients. Bull. London Math. Soc. 22 583–590.CrossRefGoogle Scholar
[2]Bateman, M. and Katz, N. H. (2012) New bounds on cap sets. J. Amer. Math. Soc. 25 585–613.CrossRefGoogle Scholar
[3]Behrend, F. A. (1946) On sets of integers which contain no three terms in arithmetic progression. Proc. Nat. Acad. Sci. 23 331–332.CrossRefGoogle Scholar
[4]Bloom, T. Translation invariant equations and the method of Sanders. Preprint: arXiv:1107.1110Google Scholar
[5]Bogolioùboff, N. (1939) Sur quelques propiétés arithmétiques des presque-périodes. Ann. Chaire Phys. Math. Kiev 4 185–205.Google Scholar
[6]Bourgain, J. (1999) On triples in arithmetic progression. Geom. Funct. Anal. 9 968–984.CrossRefGoogle Scholar
[7]Croot, E., Laba, I. and Sisask, O. Arithmetic progressions in sumsets and Lp-almost-periodicity. Preprint: arXiv:1103.6000Google Scholar
[8]Croot, E. and Sisask, O. (2010) A probabilistic technique for finding almost-periods of convolutions. Geom. Funct. Anal. 20 1367–1396.CrossRefGoogle Scholar
[9]Elkin, M. (2011) An improved construction of progression-free sets. Israel J. Math. 184 93–128.CrossRefGoogle Scholar
[10]Gowers, W. T. (1998) A new proof of Szemerédi's Theorem for arithmetic progressions of length four. Geom. Funct. Anal. 8 529–551.CrossRefGoogle Scholar
[11]Green, B. (2002) Arithmetic progressions in sumsets. Geom. Funct. Anal. 12 584–597.CrossRefGoogle Scholar
[12]Green, B. (2005) A Szemerédi-type regularity lemma in abelian groups, with applications. Geom. Funct. Anal. 15 340–376.CrossRefGoogle Scholar
[13]Green, B. and Wolf, J. (2010) A note on Elkin's improvement of Behrend's construction. In Additive Number Theory: Festschrift in Honor of the Sixtieth Birthday of Melvyn B. Nathanson, pp. 141–144.CrossRefGoogle Scholar
[14]Katz, N. H. and Koester, P. (2010) On additive doubling and energy. SIAM J. Discrete Math. 24 1684–1693.CrossRefGoogle Scholar
[15]Roth, K. F. (1953) On certain sets of integers. J.~London Math. Soc. 28 104–109.CrossRefGoogle Scholar
[16]Ruzsa, I. Z. (1991) Arithmetic progressions in sumsets. Acta Arith. 60 191–202.CrossRefGoogle Scholar
[17]Sanders, T. On the Bogolyubov–Ruzsa Lemma. Anal. PDE, to appear. arXiv:1011.0107Google Scholar
[18]Sanders, T. (2011) On Roth's Theorem on progressions. Ann. of Math. 174 619–636.CrossRefGoogle Scholar
[19]Schoen, T. and Shkredov, I. Roth's Theorem in many variables. Preprint: arXiv:1106.1601Google Scholar
[20]Tao, T. and Vu, V. (2006) Additive Combinatorics, Cambridge University Press.CrossRefGoogle Scholar