A nonconventional strong law of large numbers and fractal dimensions of some multiple recurrence sets

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 $q_1(n),...,q_\ell(n)$.