  Lacunary function systems of type $(f(M_nx))_{n\geq 1}$ for periodic
functions $f$ and sequences of fast-growing matrices $(M_n)_{n\geq 1}$ exhibit
many properties of independent random variables like satisfying the Central
Limit Theorem or the Law of the Iterated Logarithm. It is well-known that this
behaviour depends on number theoretic properties of $(M_n)_{n\geq 1}$ as well
as analytic properties of $f$. Classical techniques are essentially based on
Fourier analysis making it almost impossible to use a similar approach in the
multivariate setting. Recently Aistleitner and Berkes introduced a new method
proving the Central Limit Theorem in the one-dimensional case by approximating
$\sum_{n}f(M_nx)$ by a sum of piecewise constant periodic functions which form
a martingale differences sequence and using a Berry-Esseen type inequality.
Later this approach was used to show the Law of the Iterated Logarithm by a
consequence of Strassen's almost sure invariance principle. In this paper we
develop this method to prove the Central Limit Theorem and the Law of the
Iterated Logarithm in the multidimensional case.
