Bisector energy and pinned distances in positive characteristic

Abstract

We prove a new lower bound for the number of pinned distances over finite fields: if $A$ is a sufficiently small subset of ${\mathbb{F}}_q^2$, then there is an element in $A$ that determines $\gg |A{|}^{2/3}$ distinct distances to other elements of $A$. Combined with results for large subsets $A\subseteq {\mathbb{F}}_q^2$, this improves all previously known lower bounds on distinct distances over finite fields. In fact, we obtain an upper bound for the number of isosceles triangles determined by $A$. For that we use the concept of bisector energy. It turns out that the latter can be expressed as a point-plane incidence bound, so one can use a theorem of the third author. The conversion to this incidence problem relies on the Blaschke-Grünwald kinematic mapping -- an embedding of the group of rigid motions of ${\mathbb{F}}_q^2$ into an open subset of the projective three space. This has long been known in kinematics and geometric algebra; we provide a proof for arbitrary fields using Clifford algebras.

17 pages. Updated in response to comments by Giorgis Petridis and Thang Pham. Removed alternate proofs for existing results for large sets