Séminaire CAESAR
de combinatoire additive
Séance du 3 novembre 2011
(11 heures, Jussieu, couloir 15-25, salle 102):
Juanjo RUÉ
(Madrid)
Multilinear forms in additive combinatorics
Let A be a set of non-negative integers. Several questions
concerning additive properties of A are related to properties of the
set {a +a': a,a' in A}. However, many other interesting questions appear
when we consider the set {ka+ja' : a,a' in A}, where k and j are positive
integers. In this talk, we expose two topics related to this framework.
When A is an infinite set, we study the number of solutions of the
equation n=ka+ja', with a and a' in A and. We prove that this
function, in terms
of n, is not constant for n large enough. This partially solves a problem
posted by Sarkozy and Sós. We also show Erdós-Fuchs type
results in several cases. When A is a finite set, we study the size
of the set {ka+ja': a,a' in A} in terms of the size of A. Our results
here provide explicit constructions which are tight in some cases.
These results are based in joint works with Javier Cilleruelo (first
part) and Yahya Ould Hamidoune (second part).
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