Quantitative rational approximation on spheres

Abstract

We prove a quantitative theorem for Diophantine approximation by rational points on spheres. Our results are valid for arbitrary unimodular lattices and we further prove ‘spiraling’ results for the direction of approximates. These results are quantitative generalizations of the Khintchine-type theorem on spheres proved in Kleinbock and Merrill (Israel J Math 209:293–322, 2015).