  We provide conditions which yield a strong law of large numbers for
expressions of the form $1/N\sum_{n=1}^{N}F\big(X(q_1(n)),...,
X(q_\ell(n))\big)$ where $X(n),n\geq 0$'s is a sufficiently fast mixing vector
process with some moment conditions and stationarity properties, $F$ is a
continuous function with polinomial growth and certain regularity properties
and $q_i,i>m$ are positive functions taking on integer values on integers with
some growth conditions. Applying these results we study certain multifractal
formalism type questions concerning Hausdorff dimensions of some sets of
numbers with prescribed asymptotic frequencies of combinations of digits at
places $q_1(n),...,q_\ell(n)$.
