Circles passing through five or more integer points

Shaunna M. Plunkett-Levin¹

¹ School of Mathematics Cardiff University 23 Senghennydd Road Cardiff CF24 4AG, Wales, UK and School of Mathematics University of Bristol University Walk Bristol BS8 1TW, UK

Acta Arithmetica, Tome 158 (2013) no. 2, pp. 141-164.

Résumé

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 $2^{m-1}-1$.

Détail

DOI : 10.4064/aa158-2-3

Keywords: improvement huxley konyagins current lower bound number circles passing through five integer points conjecture improved lower bound asymptotic formula number circles passing through five integer points generalise result circles passing through five integer points giving main theorem terms cyclic polygons integer point vertices theorem geq fixed integer number cyclic polygons integer point vertices centred unit square radius leq there exists polynomial geq frac log where explicit polynomial degree m