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This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers ${\zeta }_1,{\zeta }_2,...,{\zeta }_k$. The approach relies on results on the connection between the set of all $s$-adic expansions ($s\ge 2$) of ${\zeta }_1,{\zeta }_2,...,{\zeta }_k$ and their associated approximation constants. As an application, explicit construction of real numbers ${\zeta }_1,{\zeta }_2,...,{\zeta }_k$ with prescribed approximation properties are deduced and illustrated by Matlab plots.

Following a suggestion of W.M. Schmidt and L. Summerer, we construct a proper $3$-system $(P_1,P_2,P_3)$ with the property ${\overline{\varphi }}_3=1$. In fact, our method generalizes to provide $n$-systems with ${\overline{\varphi }}_n=1$, for arbitrary $n\ge 3$. We visualize our constructions with graphics. We further present explicit examples of numbers ${\xi }_1,...,{\xi }_{n-1}$ that induce the $n$-systems in question.