Parametric approach to Khintchine's transference theorem and its generalizations

In 1926 A.Ya.Khintchine proved the famous transference inequalities connecting two dual problems. The first one concerns simultaneous approximation of given real numbers $\theta_1, \ldots, \theta_n$ by rationals, the second one concerns approximating zero with the values of the linear form $\theta_1 x_1+\ldots+\theta_n x_n +x_{n+1}$ at integer points.

In 2009 W.M.Schmidt and L.Summerer presented a new approach to Diophantine approximation, which they called parametric geometry of numbers. The formulation of Khintchine's transference theorem in terms of parametric geometry of numbers appeared to be most elegant and simple. Moreover, this approach allowed a very natural splitting of Khintchine's inequalities into a chain of inequalities between the so called intermediate Diophantine exponents.

We shall talk about application of parametric geometry of numbers to some generalizations of Khintchine's theorem. Particularly, we shall discuss transference theorems for Diophantine exponents of lattices and in Diophantine approximation with weights.