Convexity, Elementary Methods, and Distances

This paper considers an extremal version of the Erdős 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 \(d \ge 5\), then there exists \(A' \subset A\) with \(|A'| \ge |A|/2\) 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

$$\begin{aligned} | \Delta (A^{2k+3})| \le |A|^n \end{aligned}$$

then there exists \(A' \subset A\) with \(|A'| \ge |A|/2\) and

$$\begin{aligned} | kA'- kA'| \le k^2|A|^{2n-3}\log |A|. \end{aligned}$$

These results are higher dimensional analogues of a result of Hanson [4], who considered the two-dimensional case.