Let υ1, …, υd be d elements of an additive group Z (which we refer to as the steps). Consider the 2d sums ∊1υ1 + … + ∊dυd with ∊1, …, ∊d ∈ {−1, 1}. In this chapter we investigate the largest possible repetitions among these sums.

We are going to consider two, opposite, problems:

• The Littlewood–Offord problem, which is to determine, given suitable non-degeneracy conditions on υ1, …, υd and Z (e.g. excluding the trivial case when all of the steps are zero), what the largest possible repetition or concentration can occur among these sums.

• The inverse Littlewood–Offord problem, which supposes as a hypothesis that the υ1, …, υd have a large number of repeated sums, or sums concentrating in a small set, and asks what one can then deduce as a consequence on the steps υ1, …, υd.

These two problems have a similar flavor to that of sum set estimates and inverse sum set estimates respectively, and occur naturally in certain problems of additive combinatorics, in particular in considering the set of subset sums FS(A) = {Σa∈Ba : B ⊂ A} of a given set A, or in the determinant and singularity properties of random matrices with entries ±1. These problems has also arisen in several other contexts, ranging from the zeroes of complex polynomials (which was the original motivation of Littlewood and Offord [237]), to database security (see [163]).