On weighted singular vectors for multiple weights

TL;DR

This paper introduces weighted singular vectors and uniform exponents relative to multiple weights, proving invariance properties and demonstrating the existence of vectors with high exponents on certain submanifolds, with implications for divergence orbits.

Contribution

It defines weighted singular vectors and uniform exponents, proves their invariance, and extends existence results for vectors with high exponents to analytic submanifolds.

Findings

Invariance of weighted exponents for affine subspaces

Existence of totally irrational vectors with high weighted exponents

Conditions for divergence orbits in certain cones

Abstract

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.