Séminaire CAESAR
de combinatoire additive
Séance du 21 février 2013
(11 heures, UPMC Jussieu, couloir 15-25, salle 102):
Misha RUDNEV
(Université de Bristol, G. B.)
A few sum-product estimates
The Erdos-Szemerédi conjecture, or the sum-product problem,
in arithmetic combinatorics claims that whenever A is a finite
non-empty set of integers, with cardinality |A|, then
|A+A| + |A*A| > c (espilon) |A|^{2-epsilon} for any 0< epsilon < 1
and some constant c (epsilon).
The talk will review a few recent new results about sum-products over
real and complex numbers. It will show, in particular, how Solymosi's
proof of the exponent 4/3 for reals
naturally extends to the complex case, and address some technical
issues as to what can be done when
sums are replaced by differences, in which case the best known results
are obtained via a different approach, using the
Szemerédi-Trotter
theorem.
If time allows, it would also address some weaker statements of the
conjecture, where near optimal results have been recently obtained
based on the Elekes-Guth-Katz method which enabled Guth and Katz to
settle the Erdos distance problem in the plane.
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