On the Dimensional-like Characteristics Arising From Linear Inhomogeneous Approximations

TL;DR

This paper investigates the density and size of solutions to a class of inhomogeneous Diophantine approximation problems, providing asymptotic estimates for the minimal segment length containing solutions.

Contribution

It offers new asymptotic bounds for the minimal segment length in inhomogeneous Diophantine approximations, extending understanding of their dimensional characteristics.

Findings

Asymptotic estimate: L(ε) ≈ (1/ε)^{m+o(1)} as ε→0

Provides lower and upper bounds for solution density

Applicable to algebraic and badly approximable numbers

Abstract

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.