Hausdorff dimension of affine random covering sets in torus

We calculate the almost sure Hausdorff dimension of the random covering set ${\mathrm{lim\hspace{0.17em}sup}}_{n\to \mathrm{\infty }}(g_n+{\xi }_n)$ in $d$-dimensional torus ${\mathbb{T}}^d$, where the sets $g_n\subset {\mathbb{T}}^d$ are parallelepipeds, or more generally, linear images of a set with nonempty interior, and ${\xi }_n\in {\mathbb{T}}^d$ are independent and uniformly distributed random points. The dimension formula, derived from the singular values of the linear mappings, holds provided that the sequences of the singular values are decreasing.