On the Pinned Distances Problem in Positive Characteristic

Abstract

We study the Erd\H os-Falconer distance problem for a set $A\subset {\mathbb{F}}^2$, where $\mathbb{F}$ is a field of positive characteristic $p$. If $\mathbb{F}={\mathbb{F}}_p$ and the cardinality $|A|$ exceeds $p^{5/4}$, we prove that $A$ determines an asymptotically full proportion of the feasible $p$ distances. For small sets $A$, namely when $|A|\le p^{4/3}$ over any $\mathbb{F}$, we prove that either $A$ determines $\gg |A{|}^{2/3}$. For both large and small sets, the results proved are in fact for pinned distances.

A substantial revision of the older version. The main results are unchanged. arXiv admin note: text overlap with arXiv:1908.04618

References in corpus (8)

• An Improved Point-Line Incidence Bound Over Arbitrary Fields
• Distinct Distances: Open Problems and Current Bounds
• On distinct perpendicular bisectors and pinned distances in finite fields
• A point-line incidence identity in finite fields, and applications
• An example related to the Erdos-Falconer question over arbitrary finite fields
• Bisectors and pinned distances
• Kinematic Mappings for Cayley-Klein Geometries via Clifford Algebras
• A Second Wave of Expanders over Finite Fields

Cited by in corpus (1)

• Almost spanning distance trees in subsets of finite vector spaces