Diophantine approximation and special Liouville numbers

Abstract

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 ($s\geq 2$) 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.