Seminar in Algebraic Geometry and Number Theory

Abstract: A few generalizations of the classical diophantine approximation have seen considerable interest, such as approximation by restricted rationals and approximation in spaces other than Euclidean spaces. We consider a hybrid: diophantine approximation over a number field by rationals whose numerators and denominators are restricted to satisfy some congruence conditions. We will show that the adeles of a number field is a natural object in this context and the above problem can be realized as an adelic lattice point counting problem. We will discuss briefly moment formulae for adelic lattice point counting functions, obtained recently by Kim, and demonstrate how they can be utilized to obtain a quantitative result in this context. This is analogous to a result due to Schmidt and extends a result by the author, Shucheng Yu and Anish Ghosh.