Diophantine approximation on affine subspaces
In this work, we establish a clear-cut criterion for determining when an affine subspace of R^n is extremal. Specifically, we investigate the behavior of the diophantine exponent of an affine subspace and determine when it is minimal (equal to the Dirichlet exponent 1/n). Our results confirm a well-known conjecture proposed by Kleinbock. Additionally, we extend the classical theorem of Khintchine on metric Diophantine approximation to the realm of affine subspaces. A key highlight of our study is the derivation of an exact formula for the Hausdorff dimension of the set of very well approximable vectors lying on an affine subspace. Our novel estimates for the number of rational points close to an affine subspace play a crucial role in proving the above results. The counting estimate leverages Fourier analytic techniques, with a particular emphasis on the powerful large sieve inequality.