In his paper, Dirichlet gives a complete proof for n = 1 and observes that this proof can be easily extended to arbitrary values of n. Good references on this topic are Chapter II of [52] and Cassels’ book [17]. There are in the literature many papers on various generalisations of the Dirichlet Theorem and on closely related problems. A typical question asks whether for a given set of mn real n...

In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ¬ 1/x, where θ is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. F...

Let μ be a Gibbs measure of the doubling map T of the circle. For a μ-generic point x and a given sequence {rn} ⊂ R, consider the intervals (T x − rn (mod 1), T x + rn (mod 1)). In analogy to the classical Dvoretzky covering of the circle we study the covering properties of this sequence of intervals. This study is closely related to the local entropy function of the Gibbs measure and to hittin...

We demonstrate how connections between graph theory and Diophantine approximation can be used in conjunction to give simple and accessible proofs of seemingly difficult results in both subjects.

Fundamental questions in Diophantine approximation are related to the Hausdorff dimension of sets of the form {x ∈ R : δx = δ}, where δ ≥ 1 and δx is the Diophantine approximation exponent of an irrational number x. We go beyond the classical results by computing the Hausdorff dimension of the sets {x ∈ R : δx = f(x)}, where f is a continuous function. Our theorem applies to the study of the ap...