Structure of sets with small sumset
Structure theory of set addition, Astérisque no. 258  (1999), p. 77-108
@incollection{AST_1999__258__77_0,
     author = {Bilu, Yuri},
     title = {Structure of sets with small sumset},
     booktitle = {Structure theory of set addition},
     editor = {Deshouilliers Jean-Marc and Landreau Bernard and Yudin Alexander A.},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {258},
     year = {1999},
     pages = {77-108},
     zbl = {0946.11004},
     mrnumber = {1701189},
     language = {en},
     url = {http://www.numdam.org/item/AST_1999__258__77_0}
}
Bilu, Yuri. Structure of sets with small sumset, in Structure theory of set addition, Astérisque, no. 258 (1999), pp. 77-108. http://www.numdam.org/item/AST_1999__258__77_0/

[1] Bilu Y., The (α+2β)-inequality on the torus, J. London Math. Soc., to appear. | MR 1659821 | Zbl 0922.11016

[2] Bilu Y., Lev V. F., Ruzsa I. Z., Rectification principles in additive number theory, Discr. Comput. Geom., 19, 1998, 343-353. | Article | MR 1608875 | Zbl 0899.11002

[3] Bombieri E., Vaaler J., On Siegel's Lemma, Inv. Math., 73, 1983, 11-31. | Article | MR 707346 | Zbl 0533.10030

[4] Brailovski L. V, Preiman G. A., On a Product of Finite Subsets in a Torsion-Free Group, J. Algebra, 130, 1990, 462-476. | Article | MR 1051314 | Zbl 0697.20019

[5] Cassels J. W. S., An Introduction to the Geometry of Numbers, Springer, 1959. | Article | MR 157947 | Zbl 0866.11041 | Zbl 0086.26203

[6] Cauchy A. L., Recherches sur les nombres, J. École Polytech., 9, 1813, 99-116.

[7] Davenport H., On the addition of residue classes, J. London Math. Soc., 10, 1935, 30-32. | Article | JFM 61.0149.02

[8] Davenport H., A historical note, J. London Math. Soc., 22, 1947, 100-101. | Article | MR 22865 | Zbl 0029.34401

[9] Fishburn P. C., On a contribution of Freiman to additive number theory, J. Number Theory, 35, 1990, 325-334. | Article | MR 1062337 | Zbl 0703.11008

[10] Freiman G. A., Inverse problems in additive number theory VI., On the addition of finite sets III (Russian), Izv. Vyss. Učebn. Zaved. Matem., no. 3 (28), 1962, 151-157. | MR 152486

[11] Freiman G. A., Inverse problems of additive number theory, VII. On addition of finite sets, IV. The method of trigonometric sums (Russian), Izv. Vyss. Učebn. Zaved. Matem., No 3 (28), 1962, 131-144. | MR 144879 | Zbl 0171.00803

[12] Freiman G. A., Foundations of a Structural Theory of Set Addition (Russian), Kazan', 1959 ; English Translation : Translation of Mathematical Monographs 37, Amer. Math. Soc., Providence, 1973. | MR 360496 | Zbl 0271.10044

[13] Freiman G. A., What is the structure of K if K+K is small?, Number Theory, New York 1984-1985, Lecture Notes in Math., 1240, Springer, 1987, 109-134 | Article | MR 894508 | Zbl 0625.10045

[14] Freiman G. A., Heppes A., Uhrin B., A lower estimation for the cardinality of finite difference sets in n , Proc. Conf. Number Theory, Budapest 1987, Coll. Math. Soc. J. Bolyai, Budapest, 51, 1989, 125-139. | MR 1058212 | Zbl 0707.11011

[15] Hamidoune Y. O., On inverse additive problems, Preprint EC95/01, Inst. Blaise Pascal, Paris, January 1995.

[16] Hamidoune Y. O., An Isoperimetric Method in Additive Theory, J. Algebra, 179, 1996, 622-630. | Article | MR 1367866 | Zbl 0842.20029

[17] Hamidoune Y. O., Some Results in Additive Number Theory, Rapport ECH197, Univ. Paris VI, 1997. | Zbl 0985.11011

[18] Kemperman I. H. B., On small sumsets in an abelian group, Acta Math., 103, 1960, 63-88. | Article | MR 110747 | Zbl 0108.25704

[19] Kemperman I. H. B., On products of sets in a locally compact group, Fund. Math., 56, 1964, 51-68. | Article | MR 202913 | Zbl 0125.28901

[20] Kneser M., Ein Satz über abelschen Gruppen mit Anwendungen auf die Geometrie der Zahlen, Math. Z., 61, 1955, 429-434. | Article | MR 68536 | Zbl 0064.04305

[21] Kneser M., Summenmengen in lokalkompakten abelschen Gruppen, Math. Z., 66, 1956, 88-110. | Article | MR 81438 | Zbl 0073.01702

[22] Lev V. F., Structure Theorem for Multiple Addition and Frobenius Problem, J. Number Th., 58, 1996, 79-88 | Article | MR 1387726 | Zbl 0853.11017

Lev V. F., Structure Theorem for Multiple Addition and Frobenius Problem, J. Number Th., addendum : 65, 1997, 96-100. | Article | MR 1458205 | Zbl 0885.11021

[23] Lev V. F., Smeliansky P. Y., On addition of two distinct sets of integers, Acta Arith., 70, 1995, 85-91. | Article | MR 1318763 | Zbl 0817.11005

[24] Mann H., Addition Theorems : the Addition Theorems of Group Theory and Number Theory, Wiley, New York, 1965. | MR 181626 | Zbl 0127.27203

[25] Nathanson M. B., An inverse theorem for sums of sets of lattice points, J. Number Theory, 46, 1994, 29-59. | Article | MR 1268362 | Zbl 0802.11008

[26] Nathanson M. B., Additive Number Theory : 2. Inverse Theorems and the Geometry of Sumsets, Graduate Text in Math. 165, Springer, New York, 1996. | Article | MR 1477155 | Zbl 0859.11003

[27] Postnikova L. P., Fluctuations in the distribution of fractional parts (Russian), Dokl. Akad. Nauk SSSR, 161, 1965, 1282-1284 ; English Translation : Soviet Math. Dokl., 6, 1965, 597-600. | MR 181629 | Zbl 0156.05501

[28] Ruzsa I. Z., Arithmetical progressions and the number of sums, Per. Math. Hung., 25, 1992, 105-111. | Article | MR 1200845 | Zbl 0761.11005

[29] Ruzsa I. Z., A concavity property of for the measure of product sets in groups, Fund. Math., 140, 1992, 247-254. | Article | MR 1173766 | Zbl 0763.43001

[30] Ruzsa I. Z., Generalized arithmetical progressions and sumsets, Acta Math. Hungar., 65 (1994), 379-388. | Article | MR 1281447 | Zbl 0816.11008

[31] Ruzsa I. Z., Sums of sets in several dimensions, Combinatorica, 14, 1994, 485-490. | Article | MR 1312875 | Zbl 0815.11012

[32] Stanchescu Y., On addition of two distinct sets of integers, Acta Arith., 75, 1996, 191-194. | Article | MR 1379398 | Zbl 0841.11008

[33] Stanchescu Y., On the structure of sets with small doubling property on the plane (I), Acta Arith., 83, 1998, 127-141. | Article | MR 1490644 | Zbl 0885.11018

[34] Stanchescu Y., On finite difference sets, Acta Math. Hungar., 79, 1998, 123-138. | Article | MR 1611960 | Zbl 0902.05009

[35] Steinig J., On Freiman's theorems concerning the sum of two finite sets of integers, this volume. | Zbl 0962.11032

[36] Vosper A. G., The critical pairs of subsets of a group of prime order, J. London Math. Soc., 31, 1956, 200-205, 280-282. | MR 78368 | Zbl 0072.03402

[37] Zemor G., Subset sums for binary spaces, Europ. J. Combin., 13, 1992, 221-230. | Article | MR 1164767 | Zbl 0772.11004

[38] Zemor G., A generalization to noncommutative groups of a theorem of Mann, Discr. Math., 126, 1994, 365-372. | Article | MR 1264502 | Zbl 0791.05055