A probabilistic technique for finding almost-periods of convolutions

We introduce a new probabilistic technique for finding 'almost-periods' of convolutions of subsets of groups. This gives results similar to the Bogolyubov-type estimates established by Fourier analysis on abelian groups but without the need for a nice Fourier transform to exist. We also present applications, some of which are new even in the abelian setting. These include a probabilistic proof of Roth's theorem on three-term arithmetic progressions and a proof of a variant of the Bourgain-Green theorem on the existence of long arithmetic progressions in sumsets A+B that works with sparser subsets of {1, ..., N} than previously possible. In the non-abelian setting we exhibit analogues of the Bogolyubov-Freiman-Halberstam-Ruzsa-type results of additive combinatorics, showing that product sets A B C and A^2 A^{-2} are rather structured, in the sense that they contain very large iterated product sets. This is particularly so when the sets in question satisfy small-doubling conditions or high multiplicative energy conditions. We also present results on structures in product sets A B. Our results are 'local' in nature, meaning that it is not necessary for the sets under consideration to be dense in the ambient group. In particular, our results apply to finite subsets of infinite groups provided they 'interact nicely' with some other set.Comment: 29 pages, to appear in GAF

Complete Disorder is Impossible: The Mathematical Work of Walter Deuber

This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.Complete disorder is impossible – this theme of Ramsey Theory, as stated by Theodore S. Motzkin, was a guiding theme throughout Walter Deuber's scientific life.Peer Reviewe

Threshold functions and Poisson convergence for systems of equations in random sets

We present a unified framework to study threshold functions for the existence of solutions to linear systems of equations in random sets which includes arithmetic progressions, sum-free sets, $B_{h}[g]$-sets and Hilbert cubes. In particular, we show that there exists a threshold function for the property "$\mathcal{A}$ contains a non-trivial solution of $M\cdot\textbf{x}=\textbf{0}$", where $\mathcal{A}$ is a random set and each of its elements is chosen independently with the same probability from the interval of integers $\{1,\dots,n\}$. Our study contains a formal definition of trivial solutions for any combinatorial structure, extending a previous definition by Ruzsa when dealing with a single equation. Furthermore, we study the behaviour of the distribution of the number of non-trivial solutions at the threshold scale. We show that it converges to a Poisson distribution whose parameter depends on the volumes of certain convex polytopes arising from the linear system under study as well as the symmetry inherent in the structures, which we formally define and characterize.Comment: New version with minor corrections and changes in notation. 24 Page

On arithmetic structures in dense sets of integers

This dissertation deals with four problems concerning arithmetic structures in dense sets of integers. In Chapter 1 we give an exposition of the state-of-the-art technique due to Pintz, Steiger and Szemer edi which yields the best known upper bound on the density of sets whose di erence set is square-free. Inspired by the well-known fact that Fourier analysis is not su cient to detect progressions of length 4 or more, we determine in Chapter 2 a necessary and sufficient condition on a system of linear equations which guarantees the correct number of solutions in any uniform subset of Fnp. This joint work with Tim Gowers constitutes the core of this thesis and relies heavily on recent progress in so-called "quadratic Fourier analysis" pioneered by Gowers, Green and Tao. In particular, we use a structure theorem for bounded functions which provides a decomposition into a quadratically structured and a quadratically uniform part. We also present an alternative decomposition leading to improved bounds for the main result, and discuss the connections with recent results in ergodic theory. Chapter 3 deals with improved upper and lower bounds on the minimum number of monochromatic 4-term progressions in any two-colouring of ZN. Finally, in Chapter 4 we investigate the structure of the set of popular di erences of a subset of ZN. More precisely, we establish that, given a subset of size linear in N, the set of its popular differences does not always contain the complete difference set of another large set

Combinatorial and Additive Number Theory Problem Sessions: '09--'19

These notes are a summary of the problem session discussions at various CANT (Combinatorial and Additive Number Theory Conferences). Currently they include all years from 2009 through 2019 (inclusive); the goal is to supplement this file each year. These additions will include the problem session notes from that year, and occasionally discussions on progress on previous problems. If you are interested in pursuing any of these problems and want additional information as to progress, please email the author. See http://www.theoryofnumbers.com/ for the conference homepage.Comment: Version 3.4, 58 pages, 2 figures added 2019 problems on 5/31/2019, fixed a few issues from some presenters 6/29/201

The structure theory of set addition revisited

In this article we survey some of the recent developments in the structure theory of set addition.Comment: 38p