Carries, Group Theory, and Additive Combinatorics

When numbers are added in the usual way carries occur along the route. These carries cause a mess and it is natural to seek ways to minimize them. This paper proves that balanced arithmetic minimizes the proportion of carries. It also positions carries as cocycles in group theory and shows that if coset representatives for a finite-index normal subgroup H in a group G can be chosen so that the proportion of carries is less than 2/9, then there is a choice of coset representatives where no carries are needed (in other words, the extension splits). Finally, our paper makes the link between the problems above and the emerging field of additive combinatorics. Indeed the tools and techniques of this field are used in our proofs, and our examples provide an elementary introduction.