  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.
