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 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.