We prove a result on the existence of linear forms of a given Diophantine type.

Let $\theta$ be a real number with continued fraction expansion$$\theta =\left[a_0,a_1,a_2,\cdots \right],$$and let$$M=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$$be a matrix with integer entries and nonzero determinant. If $\theta$ has bounded partial quotients, then $\frac{a\theta +b}{c\theta +d}=\left[a_0^{*},a_1^{*},a_2^{*},\cdots \right]$ also has bounded partial quotients. More precisely, if $a_j\le K$ for all sufficiently large $j$, then $a_j^{*}\le |det\left(M\right)|(K+2)$ for all sufficiently large $j$. We also give a weaker bound valid for all $a_j^{*}$ with $j\ge 1$. The proofs use the homogeneous Diophantine approximation constant $L_{\infty }\left(\theta \right)={lim\; sup}_{q\to \infty }{\left(q∥q^{\theta }∥\right)}^{-1}$. We show that$$\frac{1}{\left|det\left(M\right)\right|}{L}_{\infty }\left(\theta \right)\le L_{\infty }\left(\frac{a\theta +b}{c\theta +d}\right)\le \left|det\left(M\right)\right|L_{\infty }\left(\theta \right).$$

The main result of this paper is a criterion for linear independence of continued fractions over the rational numbers. The proof is based on their special properties.

For $x\in ℂ$, $\left|x\right|\<1$, $s\in \mathbb{N}$, let ${\mathrm{Li}}_s\left(x\right)$ be the $s$-th polylogarithm of $x$. We prove that for any non-zero algebraic number $\alpha$ such that $\left|\alpha \right|\<1$, the $\mathbb{Q}\left(\alpha \right)$-vector space spanned by $1,{\mathrm{Li}}_1\left(\alpha \right),{\mathrm{Li}}_2\left(\alpha \right),\cdots$ has infinite dimension. This result extends a previous one by Rivoal for rational $\alpha$. The main tool is a method introduced by Fischler and Rivoal, which shows the coefficients of the polylogarithms in the relevant series to be the unique solution of a suitable Padé approximation problem.

Let $Q$ be a nonconstant polynomial with integer coefficients and without zeros at the non–negative integers. Essentially with the method of Hermite, a new proof is given on linear independence of values at rational points of the function$$\begin{array}{c}\hfill G\left(x\right)=\sum _{n=0}^{\infty }\frac{x^n}{Q\left(1\right)Q\left(2\right)\cdots Q\left(n\right)}.\end{array}$$

After a brief exposition of the state-of-art of research on the (Euclidean) simultaneous Diophantine approximation constants ${\theta }_s$, new lower bounds are deduced for ${\theta }_6$ and ${\theta }_7$.