Effective Diophantine Approximation over Number Fields

Abstract

Classical Diophantine approximation as first explored by Dirichlet quantifies the density of the rationals in the real line by examining how many rational numbers approximate a given real number well. This subfield of number theory has inspired many great contributions to the literature, including Minkowski’s Geometry of Numbers and Liouville’s proof of the existence of transcendental numbers. Since then, the study of Diophantine approximation has been facilitated by a number of other seemingly unrelated fields, the most successful of these arguably being the study of dynamical systems on spaces of lattices. This thesis is concerned with the counting of Diophantine approximates to linear forms by rational numbers in number fields. In it, we obtain error terms for existing asymptotic results using mean value formulas for unimodular lattices and the ergodic property of certain flows on the space comprised of these lattices. We first prove an effective ergodic theorem that associates the rate of convergence of an ergodic flow to its moments and, using existing mean value results due to Siegel and Rogers which calculate the integrals of powers of functions on the space of lattices, we provide error terms to the asymptotic count of rational approximates to real linear forms of dimension higher than 1 × 1 and describe their distribution. This work is based on an argument first given by Athreya, Ghosh and Tseng. Secondly, we describe a moment formula for lattices in number fields, generalising Rogers’ result, and apply this to obtain counting results for approximation by algebraic numbers in a given number field to linear forms, answering a question posed recently by Alam and Ghosh. We conclude by proving a new second moment formula for the primitive Siegel transform of indicator functions on a space of two dimensional lattices associated to a number field, from which we deduce appropriate error rates for totally real number fields.