Elementary methods for incidence problems in finite fields
Volume 177 / 2017
Acta Arithmetica 177 (2017), 133-142
MSC: Primary 52C10.
DOI: 10.4064/aa8225-10-2016
Published online: 22 December 2016
Abstract
We use elementary methods to prove an incidence theorem for points and spheres in $\mathbb F_q^n$. As an application, we show that any point set $P\subset \mathbb F_q^2$ with $|P|\geq 5q$ determines a positive proportion of all circles. The latter result is an analogue of Beck’s Theorem for circles which is optimal up to multiplicative constants.
