Singular Vectors and $\psi$-Dirichlet Numbers over Function Field

TL;DR

This paper explores the nature of $ ext{psi}$-Dirichlet numbers in function fields, revealing they are only rational functions, and demonstrates the existence of many irrational singular vectors with high exponents in quadratic surfaces over positive characteristic fields.

Contribution

It establishes the characterization of $ ext{psi}$-Dirichlet numbers in function fields and constructs uncountably many irrational singular vectors with large exponents.

Findings

Only rational functions are $ ext{psi}$-Dirichlet numbers in function fields.

Existence of uncountably many irrational singular vectors with large exponents.

Distinct behavior of $ ext{psi}$-Dirichlet numbers compared to real numbers.

Abstract

We show that the only $\psi$-Dirichlet numbers in a function field over a finite field are rational functions, unlike $\psi$-Dirichlet numbers in $\mathbb{R}$. We also prove that there are uncountably many totally irrational singular vectors with large uniform exponent in quadratic surfaces over a positive characteristic field.