Abstract
We generalize Khintchine's method of constructing totally irrational singular vectors and linear forms. The main result of the paper shows existence of totally irrational vectors and linear forms with large uniform Diophantine exponents on certain subsets of Double-struck capital R-n, in particular on any analytic submanifold of Double-struck capital R-n of dimension >= 2 which is not contained in a proper rational affine subspace.