  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.
