Convexity, Elementary Methods, and Distances

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.