Convexity, Elementary Methods, and Distances

TL;DR

This paper investigates the structure of point sets in high-dimensional Euclidean spaces with restricted distance sets, revealing that such sets exhibit significant additive structure, extending previous two-dimensional results.

Contribution

It generalizes Hanson’s two-dimensional results to higher dimensions, linking restricted distance growth to additive structure in point sets.

Findings

If distance set growth is limited, a large subset has small sumset.

High-dimensional analogues of Hanson’s result are established.

Restricted distance growth implies additive structure in the point set.

Abstract

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.