Combinatorial discrepancy in harmonic analysis

Given a collection of finite sets $A_1,\dots, A_n$ in $\{1,\dots,n\}$, a basic problem in combinatorial discrepancy theory is to find a colouring $f:\{1,\dots, n\}\to \{\pm1\}$ so that each sum
$$ \left|\sum_{x\in A_i} f(x) \right| $$
is as small as possible. I will discuss how the sort of combinatorial and probabilistic reasoning used to think about problems in combinatorial discrepancy can used to solve an old conjecture of J.E.Littlewood in the area of harmonic analysis.
This talk is based on joint work with Balister, Bollobás, Morris and Tiba.