When numbers are added in the usual way, 'carries' occur. Using balanced arithmetic can cut the carries down by a factor of two and nothing does better. Generalizing to other than groups, the carries are cocycles and one is led to the following problem: let H be a normal subgroup of a finite group G. Let X be coset representatives for H in G. As a measure of efficiency, let C(X) be the number of pairs x,y in X with xy in X divided by |X|^2. Thus if X can be chosen as a subgroup, C(X) = 1 and there are no carries. One of our theorems says that if C(X) is greater than 7/9 then there is a subgroup K with KH=G and K intersecting H only in the identity (so the extension splits).
The many related topics make heavy use of additive combinatorics. Our best proof of the theorem above use approximate homomorphisms. This is joint work with Fernando Shao and Kannan Soundarajan.