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Ομιλία
Plünnecke's inequality for different summands

Máté Matolcsi
Renyi Institute

Πέμπτη, 7 Αυγούστου 2008, Αίθουσα Ζ301

For finite sets of integers $A_1, A_2 \dots A_n$ we first study the cardinality of the $n$-fold sumset $S=A_1+\dots +A_n$ compared to those of $n-1$-fold sumsets $S_i=A_1+\dots +A_{i-1}+A_{i+1}+\dots A_n$. We prove a superadditivity and a submultiplicativity property for these quantities, namely:

$$(n-1)\vert S\vert \geq -1+\sum_{j=1}^n\vert S_j\vert, \ \ \ \ \ \ \mathrm{and}$$ (1)

$$\left\vert S \right\vert \leq \left( \prod _{i=1}^n \left\vert S_i \right\vert \right)^ {1\over n-1}.$$ (2)

We next prove the following version of Plünnecke's inequality for different summands: assume that for finite sets $A$, $B_1, \dots B_n$ we have information on the size of the sumsets $A+B_{i_1}+\dots +B_{i_l}$ for all choices of indices $i_1, \dots i_l.$ Then there exists a non-empty subset $X$ of $A$ such that we have 'good control' over the size of the sumset $X+B_1+\dots +B_n$. This leads us to a generalization of inequality (2).

This is joint work with K. Gyarmati and I. Z. Ruzsa.

http://www.math.uoc.gr/analysis-seminar