Some new results in quantitative Diophantine approximation

TL;DR

This paper presents two new results in quantitative Diophantine approximation, focusing on quadratic forms at prime points and inhomogeneous forms at integer points, advancing understanding of approximation properties of these forms.

Contribution

It introduces novel results on Diophantine approximation for diagonal ternary indefinite forms, specifically at prime and integer points, expanding the scope of previous research.

Findings

Quadratic forms take values at prime points with new approximation bounds.

Inhomogeneous forms of arbitrary degree are shown to have specific approximation properties.

Results contribute to the understanding of Diophantine approximation in higher dimensions.

Abstract

In this article we establish two new results on quantitative Diophantine approximation for one-parameter families of diagonal ternary indefinite forms. In the first result, we consider quadratic forms taking values at prime points. In the second, we examine inhomogeneous forms of arbitrary degree taking values at integer points.