On the statistics of the minimal solution of a linear Diophantine equation and uniform distribution of the real part of orbits in hyperbolic spaces

Abstract

Abstract. We study a variant of a problem considered by Dinaburg and Sina˘ion the statistics of the minimal solution to a linear Diophantine equation.We show that the signed ratio between the Euclidean norms of the minimalsolution and the coefficient vector is uniformly distributed modulo one. Wereduce the problem to an equidistribution theorem of Anton Good concerningthe orbits of a point in the upper half-plane under the action of a Fuchsiangroup. 1. Statement of results1.1. For a pair of coprime integers (a,b), the linear Diophantine equationax − by = 1 is well known to have infinitely many integer solutions (x,y), anytwo differing by an integer multiple of (b,a). Dinaburg and Sina˘i [2] studied thestatistics of the “minimal” such solution v ′ = (x 0 ,y 0 ) when the coefficient vec-tor v = (a,b) varies over all primitive integer vectors lying in a large box withcommensurate sides. Their notion of “minimality” was in terms of the L ∞ -norm|v ′ | ∞ := max(|x 0 |,|y 0 |), and they studied the ratio |v

Locations

• Contemporary mathematics - American Mathematical Society - View
• arXiv (Cornell University) - View - PDF