Estimate of the rate of convergence of probability distributions to a uniform distribution

The paper considers sequences of random vectors in the Euclidean space $\mathbf{R}^s (s\ge2)$: $X_1,X_2,\dots,X_n,\dots,X_n=(X_{n1},\dots,X_{ns})$, $0\le X_{nj}\le 1$, $j=1,\ldots,s$.
A deviation of a distribution of the random vectors $X_n$ from a uniform distribution on a cube $[0,1]^s$ is evaluated in terms of mathematical expectations $\mathbf{E} e^{2\pi i(m,X_n)}$, where $m$ is a vector with integer-valued coordinates. If they decrease rapidly enough as $n\to\infty$ for any convex domain $D\subset[0,1]^s$, the value $|\mathbf{P}\{X_n\in D\}-\mathrm{vol}_s(D)|$ decreases as some positive order of $1/n$.
This work is a generalization of [A. Ya. Kuznetsova and A. A. Kulikova, Moscow Univ. Comput. Math. Cybernet., 2002, no. 3, pp. 35–43], in which $s=1$ was assumed.