On weighted singular vectors for multiple weights

We introduce the notion of weighted singular vectors and weighted uniform exponent with respect to a set of weights. We prove invariance of these exponents for affine subspaces and submanifolds inside those affine subspaces. For certain analytic submanifolds, we show that there are totally irrational vectors with high weighted uniform exponent, extending the previously known existence results. Moreover, we show existence and non-existence of non-obvious divergence orbits for certain cones.