On the Dimensional-like Characteristics Arising From Linear Inhomogeneous Approximations

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.