There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area. The book covers many of the recent developments of the field, including application of important separators, branching based on linear programming, Cut & Count to obtain faster algorithms on tree decompositions, algorithms based on representative families of matroids, and use of the Strong Exponential Time Hypothesis. A number of older results are revisited and explained in a modern and didactic way. The book provides a toolbox of algorithmic techniques. Part I is an overview of basic techniques, each chapter discussing a certain algorithmic paradigm. The material covered in this part can be used for an introductory course on fixed-parameter tractability. Part II discusses more advanced and specialized algorithmic ideas, bringing the reader to the cutting edge of current research. Part III presents complexity results and lower bounds, giving negative evidence by way of W[1]-hardness, the Exponential Time Hypothesis, and kernelization lower bounds. All the results and concepts are introduced at a level accessible to graduate students and advanced undergraduate students. Every chapter is accompanied by exercises, many with hints, while the bibliographic notes point to original publications and related work.

Parameterized Algorithms

In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of Rn, and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications.

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The Brunn-Minkowski inequality

(1.1) a1x1 + . . .+ akxk = b with x1, . . . , xk in a prescribed set of integers. We saw that the vanishing of the constant term b and the sum of coefficients s = a1 + . . . + ak had a strong effect on the behaviour of equation (1.1). The condition b = 0 is equivalent to homogeneity or multiplication invariance (if x1, . . . , xk is a solution, so is tx1, . . . , txk), while s = 0 means translation invariance (if x1, . . . , xk is a solution, so is x1 + t, . . . , xk+ t). We called equations with b = s = 0 invariant , and those with b 6= 0 or s 6= 0 noninvariant . In Part I of the paper we studied invariant equations; now we treat noninvariant ones. We recall the principal notations.

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Solving a linear equation in a set of integers I

Aiming at a simultaneous extension of Khintchine’s and Furstenberg’s Recurrence theorems, we address the question if for a measure preserving system $(X,\mathcal{X},\mu,T)$
 and a set $A\in\mathcal{X}$
 of positive measure, the set of integers n such that $\mu(A{\cap} T^{n}A{\cap} T^{2n}A{\cap} \ldots{\cap} T^{kn}A)>\mu(A)^{k+1}-\epsilon$
 is syndetic. The size of this set, surprisingly enough, depends on the length (k+1) of the arithmetic progression under consideration. In an ergodic system, for k=2 and k=3, this set is syndetic, while for k≥4 it is not. The main tool is a decomposition result for the multicorrelation sequence $\int{f(x)f(T^{n}x)f(T^{2n}x){\ldots} f(T^{kn}x) \,d\mu(x)}$
 , where k and n are positive integers and f is a bounded measurable function. We also derive combinatorial consequences of these results, for example showing that for a set of integers E with upper Banach density d*(E)>0 and for all e>0, the set $$\big\{n\in\mathbb{Z}{\colon} d^*\big(E\cap(E+n)\cap(E+2n)\cap(E+3n)\big) > d^*(E)^4-\epsilon\big\}$$
 is syndetic.

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Multiple recurrence and nilsequences

Generalized arithmetical progressions and sumsets

A famous result of Freiman describes the structure of finite sets A with small doubling property. If |A + A| K|A|, then A is contained within a multidimensional arithmetic progression of dimension d(K) and size f(K)|A|. Here we prove an analogous statement valid for subsets of an arbitrary abelian group

Freiman's theorem in an arbitrary abelian group

Additive Group Theory and Non-unique Factorizations.- Notation.- Basic concepts of non-unique factorizations.- The Davenport constant and first precise arithmetical results.- The structure of sets of lengths.- Addition theorems and direct zero-sum problems.- Inverse zero-sum problems and arithmetical consequences.- Sumsets and Structure.- Notation.- Cardinality inequalities.- Structure of sets with few sums.- Location and sumsets.- Density.- Measure and topology.- Exercises.- Thematic seminars.- A survey on additive and multiplicative decompositions of sumsets and of shifted sets.- On the detailed structure of sets with small additive property.- The isoperimetric method.- Additive structure of difference sets.- The polynomial method in additive combinatorics.- Problems in additive number theory, III.- Incidences and the spectra of graphs.- Multi-dimensional inverse additive problems.

Imre Z. Ruzsa

Papers

Solving a linear equation in a set of integers I

Multiple recurrence and nilsequences

Generalized arithmetical progressions and sumsets

Freiman's theorem in an arbitrary abelian group

Combinatorial Number Theory and Additive Group Theory