*(English)*Zbl 0958.11008

In contrast to classical problems of additive representation by given numbers, say primes, “inverse” or “structural theory of set addition” asks for properties of a set if some additive characterics like the number of sums is prescribed. Though some early results can by hindsight be attributed to this field, a systematic study was initiated by the author of this paper in the sixties in a series of papers and then a monograph. This very readable survey relates the present state of the subject as seen by the most authoritative person. It also covers applications to various fields, like the behaviour of certain exponential sums, integer programming, concentration problems in probability theory, coding theory and mathematical statistics. There are no formal proofs, the author emphasizes the general framework and his philosophy instead. There are many open problems, with comments about their relevance and possible lines of attack. There is also a comprehensive bibliography.

An excellent place to get acquainted with the subject.

##### MSC:

11B13 | Additive bases, including sumsets |

11-02 | Research monographs (number theory) |

11P70 | Inverse problems of additive number theory |

11B05 | Topology etc. of sets of numbers |