Summary form only given. About two years ago Bourgain, Katz and Tao (2004) proved the following theorem, essentially stating that in every finite field, a set which does not grow much when we add all pairs of elements, and when we multiply all pairs of elements, must be very close to a subfield. Theorem 1: (Bourgain et al., 2004) For every epsi > 0 there exists a delta > 0 such that the following holds. Let F be any field with no subfield of size ges |F|epsi. For every set A sube F, with |F|epsi < |A| < |F|1 - epsi, either the sumset |A + A| > |A|1 + delta or the product set |A times A| > |A|1 + delta. This theorem revealed its fundamental nature quickly. Shortly afterwards it has found many diverse applications, including in number theory, group theory, combinatorial geometry, and the explicit construction of extractors and Ramsey graphs, mostly described in the references below. In my talk I plan to explain some of the applications, as well as to sketch the main ideas of the proof of the sum-product theorem
[1]
Avi Wigderson,et al.
2-source dispersers for sub-polynomial entropy and Ramsey graphs beating the Frankl-Wilson construction
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2006,
STOC '06.
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Avi Wigderson,et al.
Extracting randomness using few independent sources
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2004,
45th Annual IEEE Symposium on Foundations of Computer Science.
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Jean Bourgain,et al.
Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order
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2006
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Terence Tao,et al.
A sum-product estimate in finite fields, and applications
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2003,
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On the Construction of Affine Extractors
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Jean Bourgain,et al.
New results on expanders
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2006
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Guy Kindler,et al.
Simulating independence: new constructions of condensers, ramsey graphs, dispersers, and extractors
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2005,
STOC '05.