Gaps in the Spectrum of Heights of Projective Points
Number Theory
2010-03-04 v3
Abstract
Let r mod m be the least positive residue of r modulo m, and set the height of a pair (r,s) of integers, both relatively prime to m, to be the minimum over k, with 0<k<m, of (k r mod m) + (k s mod m). Denote this quantity by h(m,r,s). We give a formula for the height in terms of the continued fraction of r*s'/m, where s' is the inverse of s modulo m. Now define SPECTRUM to be the set of real numbers x with the property that there is a sequence (m_i,r_i,s_i) with m_i --> infinity, gcd(r_i,m_i)=gcd(s_i,m_i)=1, and m_i^{-1} h(m_i,r_i,s_i) --> x. The main result here is that SPECTRUM is the union of {0} and {1/k : k =1,2,...}.
Keywords
Cite
@article{arxiv.math/0703625,
title = {Gaps in the Spectrum of Heights of Projective Points},
author = {Kevin O'Bryant},
journal= {arXiv preprint arXiv:math/0703625},
year = {2010}
}
Comments
6 pages, 1 figure (newest version is shorter and quantitative)