In this paper we study multi-parameter projection theorems for fractal sets.
With the help of these estimates, we recover results about the size of $A \cdot
A+...+A \cdot A$, where $A$ is a subset of the real line of a given Hausdorff
dimension, $A+A=\{a+a': a,a' \in A \}$ and $A \cdot A=\{a \cdot a': a,a' \in
A\}$. We also use projection results and inductive arguments to show that if a
Hausdorff dimension of a subset of ${\Bbb R}^d$ is sufficiently large, then the
${k+1 \choose 2}$-dimensional Lebesgue measure of the set of $k$-simplexes
determined by this set is positive.