  The roundoff errors in computer simulations of continuous dynamical systems,
caused by finiteness of machine arithmetic, can lead to qualitative
discrepancies between phase portraits of the resulting spatially discretized
systems and the original systems. These effects can be modelled on a
multidimensional integer lattice by using a dynamical system obtained by
composing the transition operator of the original system with a quantizer. Such
models manifest pseudorandomness which can be studied using a rigorous
probability theoretic approach. To this end, the lattice $\mathbb{Z}^n$ is
endowed with a class of frequency measurable subsets and a spatial frequency
functional as a finitely additive probability measure on them. Using a
multivariate version of Weyl's equidistribution criterion, we introduce an
algebra of frequency measurable quasiperiodic subsets of the lattice. This
approach is applied to quantized linear systems with the transition operator $R
\circ L$, where $L$ is a nonsingular matrix of the original linear system in
$\mathbb{R}^n$, and the map $R$ commutes with the additive group of
translations of the lattice. For almost every $L$, the events associated with
the deviation of trajectories of the quantized and original systems are
frequency measurable quasiperiodic subsets of the lattice whose frequencies
involve geometric probabilities on finite-dimensional tori. Using the skew
products of measure preserving toral automorphisms, we prove mutual
independence and uniform distribution of the quantization errors and
investigate statistical properties of invertibility loss for the quantized
linear system, extending V.V.Voevodin's results. When $L$ is similar to an
orthogonal matrix, we establish a functional central limit theorem for the
deviations of trajectories of the quantized and original systems. These results
are demonstrated for rounded-off planar rotations.
