Construction of points realizing the regular systems of Wolfgang Schmidt and Leonard Summerer

In a series of recent papers, W. M. Schmidt and L. Summerer developed a new theory by which they recover all major generic inequalities relating exponents of Diophantine approximation to a point in $\mathbb{R}^n$, and find new ones. Given a point in $\mathbb{R}^n$, they first show how most of its exponents of Diophantine approximation can be computed in terms of the successive minima of a parametric family of convex bodies attached to that point. Then they prove that these successive minima can in turn be approximated by a certain class of functions which they call $(n,\gamma)$-systems. In this way, they bring the whole problem to the study of these functions. To complete the theory, one would like to know if, conversely, given an $(n,\gamma)$-system, there exists a point in $\mathbb{R}^n$ whose associated family of convex bodies has successive minima which approximate that function. In the present paper, we show that this is true for a class of functions which they call regular systems.