Singular Vectors and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> -Dirichlet Numbers over Function Field

We show that the only <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> -Dirichlet numbers in a function field over a finite field are rational functions, unlike <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ψ</mml:mi> </mml:math> -Dirichlet numbers in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℝ</mml:mi> </mml:math> . We also prove that there are uncountably many totally irrational singular vectors with large uniform exponent in quadratic surfaces over a positive characteristic field.