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Circles passing through five or more integer points

Volume 158 / 2013

Shaunna M. Plunkett-Levin Acta Arithmetica 158 (2013), 141-164 MSC: Primary 11P21; Secondary 11N37, 11M41. DOI: 10.4064/aa158-2-3

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

Authors

  • Shaunna M. Plunkett-LevinSchool 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
    e-mail
    e-mail

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