Additive combinatorics is the theory of counting additive structures in sets. This theory has
seen exciting developments and dramatic changes in direction in recent years thanks to its …

This book is primarily a textbook introduction to various areas of discrete geometry. In each
area, it explains several key results and methods, in an accessible and concrete manner. It …

In this paper, we prove that a set of N points in R2 has at least cNlogN distinct distances,
thus obtaining the sharp exponent in a problem of Erdős. We follow the setup of Elekes and …

Let A be a subset of a finite field F:= Z/q Z for some prime q. If| F|^ δ<| A|<| F|^ 1-δ for some
δ> 0, then we prove the estimate| A+ A|+| A ⋅ A| ≧ c (δ)| A|^ 1+ ε for some ε= ε (δ)> 0. This is …

This book demonstrates how harmonic analysis can provide penetrating insights into deep
aspects of modern analysis. It is both an introduction to the subject as a whole and an …

In the 21 st century our society is becoming more and more dependent on software systems.
The safety of these systems and the quality of our lives is increasingly dependent on the …

Twenty years ago, Ajtai, Chvátal, Newborn, Szemerédi, and, independently, Leighton
discovered that the crossing number of any graph with v vertices and e> 4 v edgesis at least …

This book explains some recent applications of the theory of polynomials and algebraic
geometry to combinatorics and other areas of mathematics. One of the first results in this …

We consider the problem of determining which point sets in some given space realise a
given distance multiset. Special cases include the “turnpike problem” where the points lie on …

Which point sets realize a given distance multiset? Interesting cases include the “turnpike
problem” where the points lie on a line, the “beltway problem” where the points lie on a loop …