  As it follows from the theory of almost periodic functions the set of integer
solutions $q$ to the Kronecker system $|\omega_{j} q - \theta_{j}| <
\varepsilon \pmod 1$, $j=1,\ldots,m$, where $1,\omega_{1},\ldots,\omega_{m}$
are linearly independent over $\mathbb{Q}$, is relatively dense in
$\mathbb{R}$. The latter means that there is $L(\varepsilon)>0$ such that any
segment of length $L(\varepsilon)$ contains at least one integer solution to
the Kronecker system. We give some lower and upper non-effective (asymptotic)
estimates for $L(\varepsilon)$ and, in particular, show that $L(\varepsilon) =
\left(\frac{1}{\varepsilon}\right)^{m+o(1)}$ as $\varepsilon \to 0$ for many
cases, including algebraic numbers as well as badly approximable numbers. We
use methods of dimension theory and Diophantine approximations of $m$-tuples
satisfying the Diophantine condition.
