Weak approximations, Diophantine exponents and two-dimensional lattices

TL;DR

This paper investigates Diophantine exponents and weak uniform approximations for two-dimensional lattices, utilizing continued fractions to develop an analog of Jarník's theory and explore inequalities between exponents.

Contribution

It introduces a two-dimensional analog of Jarník's theory, connecting Diophantine exponents with irrationality measures using continued fractions.

Findings

Established inequalities between Diophantine exponents

Linked weak approximations to irrationality measures

Extended Jarník's theory to two-dimensional lattices

Abstract

We study properties of Diophantine exponents of lattices and so-called related "weak" uniform approximations introduced in recent papers by Oleg German, in the simplest two-dimensional case. In contrast to the multidimensional case, in the two-dimensional case we can use a powerful tool of continued fractions. We develop an analog of Jarn\'{\i}k's theory dealing with inequalities between the ordinary and uniform Diophantine exponents, which turned out to be related to mutual behaviour of irrationality measure functions for two real numbers.