In this article, we prove the transcendence of certain infinite sums and products by applying the subspace theorem. In particular, we extend the results of Hančl and Rucki [‘The transcendence of certain infinite series’, Rocky Mountain J. Math.35 (2005), 531–537].

We study product theorems for matrix spaces. In particular, we prove the following theorems.

Theorem 1. For all $\varepsilon>0$, there is $\delta>0$ such that if $A\subset\mathrm{SL}_3(\mathbb{Z})$ is a finite set, then either $A$ intersects a coset of a nilpotent subgroup in a set of size at least $|A|^{1-\varepsilon}$, or $|A^3|>|A|^{1+\delta}$.

Theorem 2. Let $A$ be a finite subset of $\mathrm{SL}_2(\mathbb{C})$. Then either $A$ is contained in a virtually abelian subgroup, or $|A^3|>c|A|^{1+\delta}$ for some absolute constant $\delta>0$.

Here $A^3=\{a_1a_2a_3:a_i\in A,\ i=1,2,3\}$ is the $3$-fold product set of $A$.

For homogeneous decomposable forms F(X) in n variables with integer coefficients, we consider the number of integer solutions ${\bf x}\in\mathbb{Z}^n$ to the inequality $|F({\bf x})|\le m$ as $m\rightarrow\infty$. We give asymptotic estimates which improve on those given previously by the author in Ann. of Math. (2) 153 (2001), 767–804. Here our error terms display desirable behaviour as a function of the height whenever the degree of the form and the number of variables are relatively prime.

We present some applications of the Subspace Theorem to the investigation of the arithmetic of the values of Laurent series f(z) at S-unit points. For instance we prove that if f(qn) is an algebraic integer for infinitely many n, then h(f(qn)) must grow faster than n. By similar principles, we also prove diophantine results about power sums and transcendency results for lacunary series; these include as very special cases classical theorems of Mahler. Our arguments often appear to be independent of previous techniques in the context.