Distinct distances between a collinear set and an arbitrary set of points

Abstract

We consider the number of distinct distances between two finite sets of points in ${\mathbf{R}}^k$, for any constant dimension $k\ge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary points, such that no hyperplane orthogonal to $l$ and no hypercylinder having $l$ as its axis contains more than $O(1)$ points of $P_2$. The number of distinct distances between $P_1$ and $P_2$ is then $$\Omega \left(\min \left\{n^{2/3}{m}^{2/3},\frac{n^{10/11}{m}^{4/11}}{{\log }^{2/11}m},n^2,m^2\right\}\right).$$ Without the assumption on $P_2$, there exist sets $P_1$, $P_2$ as above, with only $O(m+n)$ distinct distances between them.

References in corpus (1)

• On the Erdos distinct distance problem in the plane