Gaps in the Spectrum of Heights of Projective Points

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&#39;/m, where s&#39; 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,...}.