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Contenuto archiviato il 2024-06-18

# Some geometric aspects of sum sets

## Final Report Summary - GEOSUMSETS (Some geometric aspects of sum sets)

The project was considering problems that line on the boundary of number theory and geometry. More precisely, the problems were from the so called additive number theory, and from the Brunn-Minkowski theory in geometry. Such problems are central in mathematics, people working in additive number theory include 2012 Abel Prize (equivalent of Nobel Prize in mathematics) laureate Szemeredi, and Fields medalist Gowers and Tao. Many of the problems in the field originate from the work of Freiman in the 1960s and 1970s. In addition people working on the Brunn-Minkowski Theory include Abel Prize laureate Gromov.

One of the central themes in additive number theory is to understand sets A in a group such that A + A has very small cardinality. The main idea is that if A + A has very small cardinality, than A has to be close being a so called arithmetic progression. We considered two variants of the problem, both of them originating in Freiman's work. First he proved that A is a subset of a torsion free abelian group, then small cardinality of A+A implies that A is contained in a relatively short arithmetic progression. In line with a conjecture of Freiman, the coordinator, the researcher and a coauthor partially extended this result to any torsion free (not only abelian) groups. The corresponding paper has been accepted for publication at the Bulletin of the London Mathematical Society.

Another classical result of Freiman is a lower estimate for A + A when A spans a d-dimensional Euclidean space. The coordinator, the researcher and a coauthor managed to characterise the equality case of the inequality. The corresponding preprint is under preparation. In these two problems of Freiman, the project addressed the more general case of the sum of two different sets, as well.

On the geometric side, the Minkowski problem in the Brunn-Minkowski theory is a classical differential equation for the so called surface area measure of a convex body, well investigated during the last hundred years. In Gromov's work arose the idea to consider the so called cone volume measure instead of the surface area measure. The researcher with coauthors managed to solve the analogue of the Minkowski problem for the cone volume measure in the case of centrally symmetric convex bodies. The corresponding paper has been accepted for publication at the Journal of the American Mathematical Society, which is one of the two top journals in mathematics.

About 80 years ago Bonnesen proved an inequality strengthening the Brunn-Minkowski inequality. The Bonnensen inequality seems to fit better with additive number theory, as earlier works of Freiman and the coordinator show. In the framework of the IEF project, the coordinator and the researcher managed to characterise the equality case of the Bonnesen inequality. The corresponding paper has been accepted for publication at the Archiv der Mathematik.

The project also considered problems linking number theory and geometry. For example, Ruzsa and Matolcsi conjectured a Brunn-Minkowski type inequality in the plane for finite sets A and B. The coordinator and the researcher verified the conjecture in many important cases, say when A consists of three points, if A and B are in convex positions, and if B is obtained from A by adding a point. The corresponding preprint is under preparation.

Another result due to a Doctor of Philosophy (PhD) student of the coordinator, the researcher and co-authors characterises hyperbolic surfaces that can be regularly triangulated. The corresponding paper has appeared in Acta. Sci. Math. (Szeged).

The project saw a rapid extension of the researcher's knowledge. Through weekly discussions with the scientist in charge and guided self-study, the researcher has learnt about fundamental methods in additive number theory, such as Szemeredi's regularity lemma, and the isoperimetric method.

The project has achieved all of its fundamental goals both in training and in research about number theory and geometry. It resulted in various publications, one of them at a top journal in mathematics. The general public has or will have open access to these publications through the website: http://arxiv.org.

While the topics of this IEF project constitute fundamental research, they are connected to Coding Theory, which applies the theory of sumsets for say write on codes and erasures, and to Computer Science, say in the case of layout arrangements and bendwidth problems. In addition, the result about the cone volume measure constitutes solving a Differential Equation of high importance. Therefore the socio-economic impact of the project will be felt only in the near future, when direct applications of the ideas might be found.