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Mathematics Department Técnico Técnico

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14/05/2013, 10:30 — 11:30 — Room P3.10, Mathematics Building
, University of Texas at Austin

Stability results for sumsets in \(\mathbb{R}^n\)

Given a Borel set \(A\) in \(\mathbb{R}^n\) of positive measure, one can consider its semisum \(S=(A+A)/2\). It is clear that \(S\) contains \(A\), and it is not difficult to prove that they have the same measure if and only if \(A\) is equal to his convex hull minus a set of measure zero. We now wonder whether this statement is stable: if the measure of \(S\) is close to the one of \(A\), is \(A\) close to his convex hull? More in general, one may consider the semisum of two different sets \(A\) and \(B\), in which case our question corresponds to proving a stability result for the Brunn-Minkowski inequality. When \(n=1\), one can approximate a set with finite unions of intervals to translate the problem onto \(\mathbb{Z}\), and in the discrete setting this question becomes a well studied problem in additive combinatorics, usually known as Freiman's Theorem. In this talk I'll review some results in the one-dimensional discrete setting, and show how to answer to this problem in arbitrary dimension.

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The Mathematics Colloquium is a series of monthly talks organized by the Department of Mathematics of IST, aiming to be a forum for the presentation of mathematical ideas or ideas about Mathematics. The Colloquium welcomes the participation of faculty, researchers and undergraduate or graduate students, of IST or other institutions, and is seen as an opportunity of bringing together and fostering the building up of ideas in an informal atmosphere.

Organizers: Conceição Amado, Lina Oliveira e Maria João Borges.