Diophantine approximation with prime denominator in real quadratic function fields

TL;DR

This paper extends Diophantine approximation results with prime denominators to real quadratic function fields, achieving an exponent of 1/4, using Vaughan's identity and Dirichlet approximation techniques.

Contribution

It provides the first analogue of Vaughan's prime approximation result for real quadratic extensions of function fields with class number one.

Findings

Proves a prime approximation exponent of 1/4 for real quadratic function fields.

Adapts Vaughan's identity and Dirichlet approximation to the function field setting.

Simplifies previous arguments for similar problems in number fields.

Abstract

In the thirties of the last century, I. M. Vinogradov proved that the inequality $||p\alpha||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent $1/5$ by $1/4$ using his celebrated identity for the von Mangoldt function and a refinement of Fourier analytic arguments. The current record is due to Matom\"aki (2009) who showed the infinitude of prime solutions of the inequality $||p\alpha||\le p^{-1/3+\varepsilon}$. This exponent $1/3$ is considered the limit of the current technology. Recently, in \cite{BaMo}, the authors established an analogue of Matom\"aki's result for imaginary quadratic extensions of the function field $k=\mathbb{F}_q(T)$. In this paper, we consider the case of real quadratic extensions of $k$ of class number 1, for which we prove a function field analogue of Vaughan's above-mentioned result (exponent $\theta=1/4$). Our method uses versions of Vaughan's identity and the Dirichlet approximation theorem for function fields. The latter was established by Arijit Ganguly in the appendix to our previous paper \cite{BaMo} on the imaginary quadratic case. We also simplify arguments in the paper \cite{BM} on the same problem for real quadratic number fields by D. Mazumder and the first-named author.