  This paper considers an extremal version of the Erd\H{o}s distinct distances
problem. For a point set $P \subset \mathbb R^d$, let $\Delta(P)$ denote the
set of all Euclidean distances determined by $P$. Our main result is the
following: if $\Delta(A^d) \ll |A|^2$ and $d \geq 5$, then there exists $A'
\subset A$ with $|A'| \geq |A|/2$ such that $|A'-A'| \ll |A| \log |A|$.
  This is one part of a more general result, which says that, if the growth of
$|\Delta(A^d)|$ is restricted, it must be the case that $A$ has some additive
structure. More specifically, for any two integers $k,n$, we have the following
information: if \[ | \Delta(A^{2k+3})| \leq |A|^n \] then there exists $A'
\subset A$ with $|A'| \geq |A|/2$ and \[ | kA'- kA'| \leq k^2|A|^{2n-3}\log|A|.
\] These results are higher dimensional analogues of a result of Hanson, who
considered the two-dimensional case.
