In this talk we are going to discuss several problems
concerning the size, density, or structure of sets of
integers with certain properties. The approaches we use
will combine ideas and tools from combinatorics, number
theory, geometry, and harmonic analysis.

Below is a list of some of the problems that we wish to address.

a) Determine the minimum density of an infinite sequence of
integers A = a1, a2, a3 ... for which the collection of
finite subset sums of A contains an infinite arithmetic
progression.

b) For a finite set A and integer k, describe the structure
of kA, the set of all sums of k distinct elements in A.

c) For a finite set A, give a "good" bound on max {|A+A|,
|AA|}, where A+A is the set of all sums of two elements in
A and AA the set of all such products.

d) For a finite set A, give a "good" bound on max |B| where
B is a subset of A for which B+B is disjoint from A.

This is joint work with B. Sudakov and V. H. Vu.