Finite groups with an automorphism cubing a large fraction of elements

We investigate the possible structures imposed on a finite group by its possession of an automorphism sending a large fraction of the group elements to their cubes, the philosophy being that this should force the group to be, in some sense, close to abelian. We prove two theorems. In the first, we completely classify all finite groups with an automorphism cubing more than half their elements. All such groups are either nilpotent class 2 or have an abelian subgroup of index at most 2. For our second theorem we show that, if a group possesses an automorphism sending more than 4/15 of its elements to their cubes, then it must be solvable. The group A_5 shows that this result is best possible. Both our main findings closely parallel results of previous authors on finite groups possessing an automorphism which inverts many group elements. The technicalities of the new proofs are somewhat more subtle, and also throw up a nice connection to a basic problem in combinatorial number theory, namely the study of subsets of finite cyclic groups which avoid non-trivial solutions to one or more translation invariant linear equations.