Singular vectors in affine subspaces and $Ψ$-Dirichlet numbers

Abstract

We prove inheritance of measure zero property of the set of singular vectors for affine subspaces and submanifolds inside them in both R and function field over finite fields. Another result of this paper shows that the only ψ-Dirichlet numbers in a function field over a finite field are rational functions, unlike ψ-Dirichlet numbers in R. We also prove that there are uncountably many totally irrational singular vectors with large uniform exponent in quadratic surfaces over a positive characteristic field.