Bisector energy and pinned distances in positive characteristic

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 $\gg |A|^{5/8}$ distinct distances to other elements of $A$. If $-1$ is not a square in $\mathbb{F}_q$, we obtain a further improvement of $\gg |A|^{13/20}$ such distances. Combined with results for large subsets $A\subseteq\mathbb{F}_q^2$, these results improve all previously known lower bounds for the number of pinned distinct distances over finite fields. These improvements are based on the point line incidence bounds of Kollar and of de Zeeuw and the third author. To apply these bounds, we embed the group of rigid motions of $\mathbb{F}_q^2$ into projective three space and into the group of affine transformations of a quadratic extension of $\mathbb{F}_q$, respectively. The proofs of our main theorem and our improvements utilise the "equivariance" of the embedding of the group of rigid motions of the plane into projective space; in fact, $\mathbb{PF}^3$ contains an open subset that is a group. This has long been known in kinematics and geometric algebra; we give a proof for arbitrary fields using Clifford algebras. We also present a new Fourier-analytic proof of the known result that any set $A \subseteq \mathbb{F}_q^2$ with $|A|\geq q^{4/3}$ yields $\gg q$ pinned distances, with close to optimal values of the implicit constant, in view of a recent counterexample by Petridis and the first author.