Topological and measure-theoretic methods in combinatorics

Often beautiful mathematics arises when abstract theories are used to understand and solve seemingly elementary problems. A prominent example of this is the mutual influence of the highly intertwined disciplines of Ramsey theory and dynamical systems. A very recent achievement is the Green Tao Theorem on the existence of arbitrarily long arithmetic progressions of primes; a seminal milestone is Furstenberg`s ergodic theoretic proof of Szemeredi`s Theorem. Another achievement closely connected with the aims of the proposed research project is the ultrafilter proof of Hindman`s theorem due to Galvin and Glazer. My own interest in Ramsey theory (in particular via the use of non-elementary methods) arose during my masters studies; I also concentrated on this topic in my Ph.D. Thesis (title: "Filters in Number Theory and Combinatorics"). Subsequently I continued research in this area in the FWF projects Probabilistic Number Theory and Analytic Combinatorics" and "Metric and Topological Aspects of Number Theoretical Problems". The proposed project would be an ideal setting to continue and finalize current research as well as to investigate certain other open questions described below. The following phenomenon in Ramsey theory is widespread: sets which are large enough to contain a certain type of highly organized structure, already contain such combinatorial structures in abundance. For instance, sets of positive density contain arithmetic progressions in high density. I am interested in analogies and subtle differences between density and partition statements in this direction. Previous related results lead to insights concerning Ramsey theory for Combined additive and multiplicative structures (partly in cooperation with V. Bergelson, N. Hindman, and D. Strauss) and further progress seems possible. Connected are certain questions concerning polynomial configurations that I want to tackle jointly with Bergelson. Another topic which will be investigated together with Bergelson concerns the Theorems of van der Waerden and Szemeredi in non-commutative groups. The corresponding recurrence results can be formulated in arbitrary groups, while the combinatorial version of Szemeredi`s Theorem is naturally stated in amenable groups. There is some evidence that a proof should be possible at least in the latter case. Sturmian sequences are defined by combinatorial properties or alternatively as coding sequences of irrational circle rotations. A generalization are coding sequences of ergodic group rotations, so-called Hartman sequences. Interestingly, (idempotent) ultrafilters can be used to gain information about the combinatorial structure of Hartman sequences, a topic which will be studied with G. Maresch, R. Winkler and C. Steineder. In the course of a recent cooperation with G. Maresch, M. Goldstern and W. Schachermayer the interplay of combinatorics and measure theoretic methods proved to be valuable in the theory of Optimal Transportation, in particular concerning the characterization of optimal transport plans. It seems promising to apply our methods to the dual part of the problem; we think that Monge-Kantorovich duality can be established under very weak assumptions.

On a very rough scale, mathematics can be divided into discrete and continuous mathematics. Discrete mathematics investigates mathematical structures of finite, countable nature. A most prominent example among the fields of discrete mathematics is the classical area of number theory. Another topic of a distinct discrete avor is combinatorics. Here a typical problem could be to count the possible results in a lottery. In contrast, continuous mathematics deals with quantities which vary "smoothly". In particular it includes topics such as analysis and calculus. To measure a certain quantity in continuous mathematics, e.g. a certain distance or area will not mean to count; rather the outcome will be an arbitrary real (decimal) number. Continuous mathematics plays a particularly prominent role in the sciences and mathematical modeling. By nature, these two branches of mathematics are in a similar contrast as the terms digital and analogue; they have distinct techniques and historically have often evolved in some independence. In this project we have been working on the connections between these two different aspects of mathematics. Ideally it is possible to translate a previously very difficult mathematical problem from one side to the other where it becomes tractable based on the different toolbox of techniques available in the other world. To mention a particular class of problems, we have been dealing with number theoretic / combinatorial questions concerning "large" sets of integers. Intuitively a set is large if it contains a positive proportion of all numbers. For instance, it is easy to agree that the set 1, 3, 5,... of odd numbers contains 50% of all numbers. However it turns out to be more complicated to formalize this for general sets and it is difficult to make universal assertions about such large sets. However, this task becomes often much more tractable if one is able to represent a set of integers through an object in the continuous world. In particular, the above set of odd numbers can be represented through an interval of length 1/2 within the unit interval [0, 1] of all numbers between 0 and 1. In the project we were able to solve several questions from number theory / combinatorics by developing corresponding counterparts in continuous mathematics. At the same time, discrete, combinatorial methods can sometimes be very successfully applied to problems that are typically considered to belong to the realm of continuous mathematics. For more information, please see: https://pf.fwf.ac.at/project_pdfs/pdf_final_reports/p21209e.pdf.