Circles passing through five or more integer points
Volume 158 / 2013
Abstract
We find an improvement to Huxley and Konyagin's current lower bound for the number of circles passing through five integer points. We conjecture that the improved lower bound is the asymptotic formula for the number of circles passing through five integer points. We generalise the result to circles passing through more than five integer points, giving the main theorem in terms of cyclic polygons with $m$ integer point vertices.
Theorem. Let $m \geq 4$ be a fixed integer. Let $W_m(R)$ be the number of cyclic polygons with $m$ integer point vertices centred in the unit square with radius $r \leq R$. There exists a polynomial $w(x)$ such that \[ W_mm \geq \frac{4^{m}}{m!}R^{2} w(\log R) (1+o(1)) \] where $w(x)$ is an explicit polynomial of degree $2^{m-1}-1$.
