Abstract: Given a sequence of positive numbers $(\ell_n)$ and a sequence of independent random variables $(\xi_n)$ uniformly distributed on the circle $S^1$, define the random covering set $E$ as follows:

E={x\in S^1 | x\in [\xi_n,\xi_n+\ell_n] for infinitely many n}=limsup[\xi_n,\xi_n+\ell_n].

The random covering set has full Lebesgue measure almost surely, precisely when $\sum_n \ell_n=\infty$. Otherwise the set is almost surely Lebesgue null.  A classical problem originally posed by Dvoretzky asks when is $E=S^1$ almost surely. A full answer was given by Shepp in 1972. For the zero Lebesgue measure case, the almost sure Hausdorff dimension of E has been found by Fan and Wu. We generalize their result by calculating the dimension of affine type random covering sets in the d-dimensional torus $T^d$.