Weak approximations, Diophantine exponents and two-dimensional lattices

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 Jarník’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.