Circles passing through five or more integer points

• Autorzy: Shaunna M. Plunkett-Levin
• Języki publikacji: EN

Abstrakty

EN

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 ≥ 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 ≤ R. There exists a polynomial w(x) such that
$W_mm ≥ (4^{m})/(m!) R^{2} w(log R)(1+o(1))$
where w(x) is an explicit polynomial of degree $2^{m-1}-1$.

Kategorie tematyczne

• 11N37: Asymptotic results on arithmetic functions
• 11M41: Other Dirichlet series and zeta functions
• 11P21: Lattice points in specified regions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2013
• Tom: 158
• Numer: 2
• Strony: 141-164

Daty

wydano

2013

Twórcy

autor

Shaunna M. Plunkett-Levin

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

Identyfikatory

DOI

10.4064/aa158-2-3