Additive Combinatorics: A Menu of Research Problems

The study of many problems in additive combinatorics, such as Szemer\'edi's theorem on arithmetic progressions, is made easier by first studying models for the problem in F_p^n for some fixed small prime p. We give a number of examples of finite field models of this type, which allows us to introduce some of the central ideas in additive combinatorics relatively cleanly. We also give an indication of how the intuition gained from the study of finite field models can be helpfu...

One of the great pleasures of working with Imre Leader is to experience his infectious delight on encountering a compelling combinatorial problem. This collection of open problems in combinatorics has been put together by a subset of his former PhD students and students-of-students for the occasion of his 60th birthday. All of the contributors have been influenced (directly or indirectly) by Imre: his personality, enthusiasm and his approach to mathematics. The problems inclu...

The main purpose of this survey is to introduce an inexperienced reader to additive prime number theory and some related branches of analytic number theory. We state the main problems in the field, sketch their history and the basic machinery used to study them, and try to give a representative sample of the directions of current research.

This is an expanded version of the Notices of the AMS column with the same title. The text is unchanged, but we added acknowledgements and a large number of endnotes which provide the context and the references.

Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry is often concerned with the problem of bounding the behaviour of arbitrary finite collections of geometric objects such as points, lines, or circles with respect to geometric operations such as incidence or distance. Given the presence of ar...

The present work has been designed for students in secondary school and their teachers in mathematics. We will show how with the help of our knowledge of number systems we can solve problems from other fields of mathematics for example in combinatorial analysis and most of all when proving some combinatorial identities. To demonstrate discussed in this article method we have chosen several suitable mathematical tasks.

A survey paper on some recent results on additive problems with prime powers.

Let $A_1,\ldots,A_n$ be finite subsets of an additive abelian group $G$ with $|A_1|=\cdots=|A_n|\ge2$. Concerning the two new kinds of restricted sumsets $$L(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n,\ \text{and}\ a_i\not=a_{i+1} \ \text{for}\ 1\le i<n\}$$ and $$C(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_i\in A_i\ (1\le i\le n),\ \text{and}\ a_i\not=a_{i+1} \ \text{for}\ 1\le i<n,\ \text{and}\ a_n\not=a_1\}$$ recently introduced by the second author, when $G...

We introduce a geometric method to study additive combinatorial problems. Using equivariant cohomology we reprove the Dias da Silva-Hamidoune theorem. We improve a result of Sun on the linear extension of the Erd\H{o}s-Heilbronn conjecture. We generalize a theorem of G. K\'os (the Grashopper problem) which in some sense is a simultaneous generalization of the Erd\H{o}s-Heilbronn conjecture. We also prove a signed version of the Erd\H{o}s-Heilbronn conjecture and the Grashoppe...

This paper collects some problems that I have encountered during the years, have puzzled me and which, to the best of my knowledge, are still open. Most of them are well-known and have been first stated by other authors. In this sad season of lockdown, I modestly try to contribute to scientific interaction at a distance. Therefore all comments and exchange of information are most welcome.