|
The structure theory of set addition revisited
Author:
Tom Sanders
Journal:
Bull. Amer. Math. Soc. 50 (2013), 93-127
MSC (2010):
Primary 11B13
Posted:
October 2, 2012
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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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Yuri
Bilu, Structure of sets with small sumset, Astérisque
258 (1999), xi, 77–108 (English, with English and
French summaries). Structure theory of set addition. MR 1701189
(2000h:11109)
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N.
Bogolioùboff, Sur quelques propriétés
arithmétiques des presque-périodes, Ann. Chaire Phys.
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Jean
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Arithmetic progressions in sumsets and -almost-periodicity. 2011, arXiv:1103.6000.
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Ernie
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Z. Ruzsa, Integer sum sets containing long arithmetic
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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.
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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.
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B.
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Konyagin and Izabella
Łaba, Distance sets of well-distributed planar sets for
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Martin
Kneser, Abschätzung der asymptotischen Dichte von
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(German). MR
0056632 (15,104c)
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S.
V. Konyagin, On Freiman’s theorem in finite fields, Mat.
Zametki 84 (2008), no. 3, 472–474 (Russian);
English transl., Mat. Zametki 84 (2008), no. 4,
435–438. MR 2473762
(2009i:11013)
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S. Lovett.
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Jorge
M. López and Kenneth
A. Ross, Sidon sets, Marcel Dekker Inc., New York, 1975.
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Giorgis
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Walter
Rudin, Trigonometric series with gaps, J. Math. Mech.
9 (1960), 203–227. MR 0116177
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Walter
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)
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I.
Z. Ruzsa, On the cardinality of 𝐴+𝐴 and
𝐴-𝐴, Combinatorics (Proc. Fifth Hungarian Colloq.,
Keszthely, 1976) Colloq. Math. Soc. János Bolyai, vol. 18,
North-Holland, Amsterdam, 1978, pp. 933–938. MR 519317
(80c:05016)
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Imre
Z. Ruzsa, An application of graph theory to additive number
theory, Sci. Ser. A Math. Sci. (N.S.) 3 (1989),
97–109. MR
2314377
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I.
Z. Ruzsa, Generalized arithmetical progressions and sumsets,
Acta Math. Hungar. 65 (1994), no. 4, 379–388.
MR
1281447 (95k:11011), http://dx.doi.org/10.1007/BF01876039
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Imre
Z. Ruzsa, An analog of Freiman’s theorem in groups,
Astérisque 258 (1999), xv, 323–326 (English,
with English and French summaries). Structure theory of set addition. MR 1701207
(2000h:11111)
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Alex
Samorodnitsky, Low-degree tests at large distances,
STOC’07—Proceedings of the 39th Annual ACM Symposium on Theory
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E.
Szemerédi and V.
Vu, Long arithmetic progressions in
sumsets: thresholds and bounds, J. Amer. Math.
Soc. 19 (2006), no. 1, 119–169. MR 2169044
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E.
Szemerédi, On sets of integers containing no four elements
in arithmetic progression, Acta Math. Acad. Sci. Hungar.
20 (1969), 89–104. MR 0245555
(39 #6861)
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E.
Szemerédi, On sets of integers containing no 𝑘
elements in arithmetic progression, Acta Arith. 27
(1975), 199–245. Collection of articles in memory of Juriĭ
Vladimirovič Linnik. MR 0369312
(51 #5547)
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Terence
Tao, Product set estimates for non-commutative groups,
Combinatorica 28 (2008), no. 5, 547–594. MR 2501249
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Terence
Tao, Freiman’s theorem for solvable groups, Contrib.
Discrete Math. 5 (2010), no. 2, 137–184. MR 2791295
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Terence
Tao and Van
Vu, Additive combinatorics, Cambridge Studies in Advanced
Mathematics, vol. 105, Cambridge University Press, Cambridge, 2006. MR 2289012
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Terence
Tao and Van
Vu, On the singularity probability of
random Bernoulli matrices, J. Amer. Math.
Soc. 20 (2007), no. 3, 603–628. MR 2291914
(2008h:60027), http://dx.doi.org/10.1090/S0894-0347-07-00555-3
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J. Wolf.
A local inverse theorem in . Preprint, 2009.
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C. E. Zohar.
On sums of generating sets in . 2011, arXiv:1108.4902.
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- 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 -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 . 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 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 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 and . 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 . 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 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 . Preprint, 2009.
- [Zoh11]
- C. E. Zohar.
On sums of generating sets in . 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
Email:
tom.sanders@maths.ox.ac.uk
DOI:
http://dx.doi.org/10.1090/S0273-0979-2012-01392-7
PII:
S 0273-0979(2012)01392-7
Received by editor(s):
July 16, 2012
Received by editor(s) in revised form:
August 20, 2012
Posted:
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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