Structure theory of set addition.

*(English)*Zbl 0919.00044
Astérisque. 258. Paris: Société Mathématique de France, 436 p. (1999).

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For a long time, additive number theory, motivated by conjectures such as those of Goldbach or Waring, has been concerned by the study of additive properties of special sequences. In the 1930’s it was noticed that the consideration of the additive properties of general sequences turned out, not only to be a beautiful subject for its own sake, but was able to lead to improvements in the study of special sequences: thus, in the paper founding this philosophy, Schnirel’man introduced a density on sets of integers, gave a general lower bound for the density of the sum of two sets, and applied it to the special sequence of primes to show that every integer can be written as a sum of a uniformly bounded number of primes. Additive number theory evolved towards the definition of invariants for sets of (non-necessarily commutative) monoids and the study of the invariants for the “sum” of different sets in terms of the invariants of those sets.

A new trend appeared in the 1950’s, with authors like M. Kneser and G. A. Freiman, which is sometimes described as inverse additive theory: knowing that the relation between the invariants of a family of sets and the invariant of their sum is extremal (or close to), what can be said on the structure of the sets themselves?

In recent years, there has been a renewed interest for this approach which turns out to have applications to several other fields. It has seemed appropriate to gather in a single volume 24 contemporary original research papers and 3 survey articles dealing with the structure theory of set addition and its applications to elementary or combinatorial number theory, group theory, integer programming and probability theory.

Indexed articles:

Freiman, Gregory A., Structure theory of set addition, 1-33 [Zbl 0958.11008]

Besser, Amnon, Sets of integers with large trigonometric sums, 35-76 [Zbl 0958.11055]

Bilu, Yuri, Structure of sets with small sumset, 77-108 [Zbl 0946.11004]

Sárkőzy, András, On finite addition theorems, 109-127 [Zbl 0969.11003]

Steinig, John, On Freiman’s theorems concerning the sum of two finite sets of integers, 129-140 [Zbl 0962.11032]

Deshouillers, Jean-Marc; Freiman, Gregory A., On an additive problem of Erdős and Straus. II, 141-148 [Zbl 0979.11005]

Deshouillers, Jean-Marc; Freiman, Gregory A.; Sós, Vera; Temkin, Mikhail, On the structure of sum-free sets. II, 149-161 [Zbl 0964.11006]

Freiman, Gregory A.; Low, Lewis; Pitman, Jane, Sumsets with distinct summands and the Erdős-Heilbronn conjecture on sums of residues, 163-172 [Zbl 0948.11008]

Hennecart, François; Robert, Gilles; Yudin, Alexander, On the volume of sums and differences, 173-178 [Zbl 0969.11011]

Lev, Vsevolod F., The structure of multisets with a small volume of subset sums, 179-186 [Zbl 0969.11005]

Lipkin, Edith, Subset sums of sets of residues, 187-193 [Zbl 0946.11005]

Nathanson, Melvyn B.; Tenenbaum, Gérald, Inverse theorems and the number of sums and products, 195-204 [Zbl 0947.11008]

Nicolas, Jean-Louis, Stratified sets, 205-215 [Zbl 0978.11055]

Stanchescu, Yonutz V., On the structure of sets of lattice points in the plane with a small doubling property, 217-240 [Zbl 0979.11018]

Berkovich, Yakov, Non-solvable groups with a large fraction of involutions, 241-248 [Zbl 0945.20009]

Berkovich, Yakov, Questions on set squaring in groups, 249-253 [Zbl 0945.20015]

Brodsky, Sergei, On groups generated by a pair of elements with small third or fourth power, 255-279 [Zbl 0944.20018]

Hamidoune, Yahya Ould, On small subset product in a group, 281-308 [Zbl 0945.20011]

Herzog, Marcel, New results on subset multiplication in groups, 309-315 [Zbl 0944.20019]

Vsevolod, Lev F., On small sumsets in abelian groups, 317-321 [Zbl 0956.11008]

Ruzsa, Imre Z., An analog of Freiman’s theorem in groups, 323-326 [Zbl 0946.11007]

Cohen, Gérard; Zémor, Gilles, Subset sums and coding theory., 327-339 [Zbl 1044.94016]

Chaimovich, Mark, New structural approach to integer programming: a survey, 341-362 [Zbl 0987.90060]

Chaimovich, Mark, New algorithm for dense subset-sum problem, 363-373 [Zbl 0987.90061]

Plagne, Alain, On the two-dimensional subset sum problem, 375-409 [Zbl 0947.11013]

Deshouillers, Jean-Marc; Freiman, Gregory A.; Moran, William, On series of discrete random variables. I: Real trinomial distributions with fixed probabilities, 411-423 [Zbl 0945.60011]

Deshouillers, Jean-Marc; Freiman, Gregory A.; Yudin, Alexander A., On bounds for the concentration function. I, 425-436 [Zbl 0944.60028]

##### MSC:

00B25 | Proceedings of conferences of miscellaneous specific interest |

11-06 | Proceedings, conferences, collections, etc. pertaining to number theory |

05-06 | Proceedings, conferences, collections, etc. pertaining to combinatorics |

05A17 | Combinatorial aspects of partitions of integers |

05D05 | Extremal set theory |

11B05 | Density, gaps, topology |

11B13 | Additive bases, including sumsets |

11B25 | Arithmetic progressions |

11B75 | Other combinatorial number theory |

11P99 | Additive number theory; partitions |

20E34 | General structure theorems for groups |