Scientific Calendar Event



Starts 19 Apr 2004 16:00
Ends 19 Apr 2004 20:00
Central European Time
ICTP
Main Building Seminar Room
Strada Costiera, 11 I - 34151 Trieste (Italy)
Abstract: Additive number theory studies the connection between a family of integral sequences and their sum, i.e. the set of those integers which can be represented as sums of elements from the given family of sequences. In "direct problems", the initial sequences are known and the question is to get some knowledge on their sum Due to their history, "special direct problems" are the best known, for example, Goldbach's problem (1742), not yet solved (worth one million dollars), stating that every even integer larger than 2 is a sum of 2 primes, or Waring's problem (1770), stating that every positive integer is a sum of four squares (indeed solved during that same year 1770 by Lagrange), of nine cubes (solved in 1909-1912 by Wieferich and Kemperman), of nineteen fourth powers (solved in 1985 by Balasubramanian, Deshouillers and Dress). L. Shnirel'man introduced "general direct problems" in 1930 when he showed that if a sequence is sufficiently rich (if it has a positive density), then every integer is a sum of a bounded number of elements from this set. By this approach, joined to a "sieve" argument, he gave a partial answer to Goldbach's problem and showed that every integer can be expressed as a sum of a (uniformly) bounded number of primes. The works of Freiman and Kneser, in the 1950's, opened the road to "inverse problems", where one tries and get some knowledge on the summands from a knowledge on their sum (an easy and basic example is the fact that if a finite set of integer with $k$ elements is doubled and leads to a sum-set with $2k-1$ elements, then the initial set is an arithmetic progression [as one easily checks, the converse statement is obvious]). This colloquium talk, for which no special knowledge is required, is a guided tour through additive number theory, its result, its method and its applications.
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