Distinct distances between points and lines in 𝔽 q 2

Abstract

Abstract We prove that if we are given a set of points {\mathcal{P}} and set of lines {\mathcal{L}} in {\mathbb{F}_{q}^{2}} such that {|\mathcal{P}||\mathcal{L}|\gtrsim q^{8/3}} , then the set of distinct distances between points from {\mathcal{P}} and lines from {\mathcal{L}} contains a positive proportion of all distances. Using the same techniques, we also obtain a generalization of this result in higher dimensional cases.