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The structure theory of set addition revisited

Author: Tom Sanders
Journal: Bull. Amer. Math. Soc. 50 (2013), 93-127
MSC (2010): Primary 11B13
Published electronically: October 2, 2012
MathSciNet review: 2994996
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Abstract: In this article we survey some of the recent developments in the structure theory of set addition.

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  • [Bil99] Y. Bilu.
    Structure of sets with small sumset.
    Astérisque, (258):xi, 77-108, 1999.
    Structure theory of set addition. MR 1701189 (2000h:11109)
  • [Bog39] N. Bogolioùboff.
    Sur quelques propriétés arithmétiques des presque-périodes.
    Ann. Chaire Phys. Math. Kiev, 4:185-205, 1939. MR 0020164 (8:512b)
  • [Bou99] J. Bourgain.
    On triples in arithmetic progression.
    Geom. Funct. Anal., 9(5):968-984, 1999. MR 1726234 (2001h:11132)
  • [Bou08] J. Bourgain.
    Roth's theorem on progressions revisited.
    J. Anal. Math., 104:155-192, 2008. MR 2403433 (2009g:11011)
  • [BS94] A. Balog and E. Szemerédi.
    A statistical theorem of set addition.
    Combinatorica, 14(3):263-268, 1994. MR 1305895 (95m:11019)
  • [Cha02] M.-C. Chang.
    A polynomial bound in Freĭman's theorem.
    Duke Math. J., 113(3):399-419, 2002. MR 1909605 (2003d:11151)
  • [Cha04] M.-C. Chang.
    On problems of Erdős and Rudin.
    J. Funct. Anal., 207(2):444-460, 2004. MR 2032997 (2004j:11022)
  • [Cha09] M.-C. Chang.
    Some consequences of the polynomial Freĭman-Ruzsa conjecture.
    C. R. Math. Acad. Sci. Paris, 347(11-12):583-588, 2009. MR 2532910 (2010e:11005)
  • [CŁS11] E. S. Croot, I. Łaba, and O. Sisask.
    Arithmetic progressions in sumsets and $ {L}^p$-almost-periodicity.
    2011, arXiv:1103.6000.
  • [CS10] E. S. Croot and O. Sisask.
    A probabilistic technique for finding almost-periods of convolutions.
    Geom. Funct. Anal., 20(6):1367-1396, 2010. MR 2738997 (2012d:11019)
  • [DHP04] J.-M. Deshouillers, F. Hennecart, and A. Plagne.
    On small sumsets in $ (\mathbb{Z}/2\mathbb{Z})^n$.
    Combinatorica, 24(1):53-68, 2004. MR 2057683 (2005f:11231)
  • [FHR92] G. A. Freĭman, H. Halberstam, and I. Z. Ruzsa.
    Integer sum sets containing long arithmetic progressions.
    J. London Math. Soc. (2), 46(2):193-201, 1992. MR 1182477 (93j:11008)
  • [Fre66] G. A. Freĭman.
    Nachala strukturnoi teorii slozheniya mnozhestv.
    Kazan. Gosudarstv. Ped. Inst, 1966. MR 0360495 (50:12943)
  • [Fre73a] G. A. Freĭman.
    Foundations of a structural theory of set addition.
    American Mathematical Society, Providence, R. I., 1973.
    Translated from the Russian, Translations of Mathematical Monographs, Vol 37.
  • [Fre73b] G. A. Freĭman.
    Groups and the inverse problems of additive number theory.
    In Number-theoretic studies in the Markov spectrum and in the structural theory of set addition (Russian), pages 175-183. Kalinin. Gos. Univ., Moscow, 1973.
  • [Gow98] W. T. Gowers.
    A new proof of Szemerédi's theorem for arithmetic progressions of length four.
    Geom. Funct. Anal., 8(3):529-551, 1998. MR 1631259 (2000d:11019)
  • [Gow01] W. T. Gowers.
    A new proof of Szemerédi's theorem.
    Geom. Funct. Anal., 11(3):465-588, 2001. MR 1844079 (2002k:11014)
  • [GR06] B. J. Green and I. Z. Ruzsa.
    Sets with small sumset and rectification.
    Bull. London Math. Soc., 38(1):43-52, 2006. MR 2201602 (2006i:11027)
  • [GR07] B. J. Green and I. Z. Ruzsa.
    Freĭman's theorem in an arbitrary abelian group.
    J. Lond. Math. Soc. (2), 75(1):163-175, 2007. MR 2302736 (2007m:20087)
  • [Gre02] B. J. Green.
    Arithmetic progressions in sumsets.
    Geom. Funct. Anal., 12(3):584-597, 2002. MR 1924373 (2003i:11148)
  • [GS08] B. J. Green and T. Sanders.
    A quantitative version of the idempotent theorem in harmonic analysis.
    Ann. of Math. (2), 168(3):1025-1054, 2008, arXiv:math/0611286. MR 2456890 (2010c:11013)
  • [GT06] B. J. Green and T. C. Tao.
    Compressions, convex geometry and the Freĭman-Bilu theorem.
    Q. J. Math., 57(4):495-504, 2006. MR 2277597 (2007g:11013)
  • [GT08] B. J. Green and T. C. Tao.
    An inverse theorem for the Gowers $ U\sp 3(G)$ norm.
    Proc. Edinb. Math. Soc. (2), 51(1):73-153, 2008. MR 2391635 (2009g:11012)
  • [GT09a] B. J. Green and T. C. Tao.
    Freĭman's theorem in finite fields via extremal set theory.
    Combin. Probab. Comput., 18(3):335-355, 2009. MR 2501431 (2010f:11176)
  • [GT09b] B. J. Green and T. C. Tao.
    A note on the Freĭman and Balog-Szemerédi-Gowers theorems in finite fields.
    J. Aust. Math. Soc., 86(1):61-74, 2009. MR 2495998 (2010d:11015)
  • [GT10] B. J. Green and T. C. Tao.
    An equivalence between inverse sumset theorems and inverse conjectures for the $ U^3$ norm.
    Math. Proc. Cambridge Philos. Soc., 149(1):1-19, 2010. MR 2651575 (2011g:11019)
  • [JL01] W. B. Johnson and J. Lindenstrauss, editors.
    Handbook of the geometry of Banach spaces. Vol. I.
    North-Holland Publishing Co., Amsterdam, 2001. MR 1863689 (2003f:46013)
  • [KK10] N. H. Katz and P. Koester.
    On additive doubling and energy.
    SIAM J. Discrete Math., 24(4):1684-1693, 2010. MR 2746716 (2012d:11020)
  • [KŁ06] S. V. Konyagin and I. Łaba.
    Distance sets of well-distributed planar sets for polygonal norms.
    Israel J. Math., 152:157-179, 2006. MR 2214458 (2006m:11032)
  • [Kne53] M. Kneser.
    Abscätzungen der symptoticschen dichte von summenmengen.
    Math. Z., 58:459-484, 1953. MR 0056632 (15:104c)
  • [Kon08] S. V. Konyagin.
    On Freĭman's theorem in finite fields.
    Mat. Zametki, 84(3):472-474, 2008. MR 2473762 (2009i:11013)
  • [Lov10] S. Lovett.
    Equivalence of polynomial conjectures in additive combinatorics.
    2010, arXiv:1001.3356.
  • [LR75] J. M. López and K. A. Ross.
    Sidon sets.
    Marcel Dekker Inc., New York, 1975.
    Lecture Notes in Pure and Applied Mathematics, Vol. 13. MR 0440298 (55:13173)
  • [Pet11a] G. Petridis.
    New proofs of Plünnecke-type estimates for product sets in groups.
    2011, arXiv:1101.3507.
  • [Pet11b] G. Petridis.
    Plünnecke's inequality.
    2011, arXiv:1101.2532. MR 2847275
  • [Plü69] H. Plünnecke.
    Eigenschaften und Abschätzungen von Wirkungsfunktionen.
    BMwF-GMD-22. Gesellschaft für Mathematik und Datenverarbeitung, Bonn, 1969. MR 0252348 (40:5569)
  • [Rud60] W. Rudin.
    Trigonometric series with gaps.
    J. Math. Mech., 9:203-227, 1960. MR 0116177 (22:6972)
  • [Rud90] W. Rudin.
    Fourier analysis on groups.
    Wiley Classics Library. John Wiley & Sons Inc., New York, 1990.
    Reprint of the 1962 original, A Wiley-Interscience Publication. MR 1038803 (91b:43002)
  • [Ruz78] I. Z. Ruzsa.
    On the cardinality of $ A+A$ and $ A-A$.
    In Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, volume 18 of Colloq. Math. Soc. János Bolyai, pages 933-938. North-Holland, Amsterdam, 1978. MR 519317 (80c:05016)
  • [Ruz89] I. Z. Ruzsa.
    An application of graph theory to additive number theory.
    Scientia, Ser. A., 3:97-109, 1989. MR 2314377
  • [Ruz94] I. Z. Ruzsa.
    Generalized arithmetical progressions and sumsets.
    Acta Math. Hungar., 65(4):379-388, 1994. MR 1281447 (95k:11011)
  • [Ruz99] I. Z. Ruzsa.
    An analog of Freĭman's theorem in groups.
    Astérisque, (258):xv, 323-326, 1999.
    Structure theory of set addition. MR 1701207 (2000h:11111)
  • [Sam07] A. Samorodnitsky.
    Low-degree tests at large distances.
    In STOC '07--Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pages 506-515. ACM, New York, 2007. MR 2402476 (2009f:68077)
  • [San10] T. Sanders.
    On the Bogolyubov-Ruzsa lemma.
    Anal. PDE, to appear, 2010, arXiv:1011.0107.
  • [Sch03] T. Schoen.
    Multiple set addition in $ \mathbb{Z}\sb p$.
    Integers, 3:A17, 6 pp. (electronic), 2003. MR 2036483 (2004j:11012)
  • [Sch11] T. Schoen.
    Near optimal bounds in Freĭman's theorem.
    Duke Math. J., 158:1-12, 2011. MR 2794366 (2012f:11018)
  • [SV06] E. Szemerédi and V. Vu.
    Long arithmetic progressions in sumsets: thresholds and bounds.
    J. Amer. Math. Soc., 19(1):119-169, 2006. MR 2169044 (2006j:11015)
  • [Sze69] E. Szemerédi.
    On sets of integers containing no four elements in arithmetic progression.
    Acta Math. Acad. Sci. Hungar., 20:89-104, 1969. MR 0245555 (39:6861)
  • [Sze75] E. Szemerédi.
    On sets of integers containing no $ k$ elements in arithmetic progression.
    Acta Arith., 27:199-245, 1975.
    Collection of articles in memory of Juriĭ Vladimirovič Linnik. MR 0369312 (51:5547)
  • [Tao08] T. C. Tao.
    Product set estimates for non-commutative groups.
    Combinatorica, 28(5):547-594, 2008. MR 2501249 (2010b:11017)
  • [Tao10] T. C. Tao.
    Freĭman's theorem for solvable groups.
    Contrib. Disc. Math., 5(2):137-184, 2010. MR 2791295 (2012e:05063)
  • [TV06] T. C. Tao and H. V. Vu.
    Additive combinatorics, volume 105 of Cambridge Studies in Advanced Mathematics.
    Cambridge University Press, Cambridge, 2006. MR 2289012 (2008a:11002)
  • [TV07] T. C. Tao and V. H. Vu.
    On the singularity probability of random Bernoulli matrices.
    J. Amer. Math. Soc., 20(3):603-628 (electronic), 2007. MR 2291914 (2008h:60027)
  • [Wol09] J. Wolf.
    A local inverse theorem in $ \mathbb{F}_2^n$.
    Preprint, 2009.
  • [Zoh11] C. E. Zohar.
    On sums of generating sets in $ \mathbb{Z}_2^n$.
    2011, arXiv:1108.4902.

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Additional Information

Tom Sanders
Affiliation: Mathematical Institute, University of Oxford, 24-29 St. Giles’, Oxford OX1 3LB, England

Received by editor(s): July 16, 2012
Received by editor(s) in revised form: August 20, 2012
Published electronically: October 2, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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