Circles passing through five or more integer points

Shaunna M. Plunkett-Levin. "Circles passing through five or more integer points." Acta Arithmetica 158.2 (2013): 141-164. <http://eudml.org/doc/278913>.

@article{ShaunnaM2013,
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 ≥ 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 $2^\{m-1\}-1$.},
author = {Shaunna M. Plunkett-Levin},
journal = {Acta Arithmetica},
keywords = {integer points; cyclic polygons; circles},
language = {eng},
number = {2},
pages = {141-164},
title = {Circles passing through five or more integer points},
url = {http://eudml.org/doc/278913},
volume = {158},
year = {2013},
}

TY - JOUR
AU - Shaunna M. Plunkett-Levin
TI - Circles passing through five or more integer points
JO - Acta Arithmetica
PY - 2013
VL - 158
IS - 2
SP - 141
EP - 164
AB - 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 $2^{m-1}-1$.
LA - eng
KW - integer points; cyclic polygons; circles
UR - http://eudml.org/doc/278913
ER -