Singularity, weighted uniform approximation, intersections and rates

Abstract

A classical argument was introduced by Khintchine in 1926 in order to exhibit the existence of totally irrational singular linear forms in two variables. This argument was subsequently revisited and extended by many authors. In the present paper we adapt Khintchine's argument to show that the sets of very singular matrices and their weighted analogues intersect many manifolds and fractals, and have strong intersection properties. We also obtain new bounds on the rate of singularity which can be attained by column vectors in analytic submanifolds of dimension at least 2 in n-dimensional space.