The purpose of this paper is to use the filtration that appeared in Ru and Vojta [Amer. J. Math. 142 (2020), pp. 957-991] extend result Blum-Jonsson [Adv. 365 p. 57], as well explore some connections between notion K <mml:annotation encoding="a...

In the present survey paper, we explain how the theory of Hausdorff dimension and Hausdorff measure is used to answer certain questions in Diophantine approximation. The final section is devoted to a discussion around the Diophantine properties of the points lying in the middle third Cantor set.

Abstract We improve on a result by Svetlana Jitomirskaya and Wencai Liu dealing with inhomogeneous Diophantine approximation in the coprime setting.

The well known theorems of Khintchine and Jarník in metric Diophantine approximation provide a comprehensive description of the measure theoretic properties of real numbers approximable by rational numbers with a given error. Various generalisations of these fundamental results have been obtained for other settings, in particular, for curves and more generally manifolds. In this paper we develo...

In this paper we consider a Schrödinger eigenvalue problem with a potential consisting of a periodic part together with a compactly supported defect potential. Such problems arise as models in condensed matter to describe color in crystals as well as in engineering to describe optical photonic structures. We are interested in studying the existence of point eigenvalues in gaps in the essential ...

We compute the Hausdorff dimension of sets of very well approximable vectors on rational quadrics. We use ubiquitous systems and the geometry of locally symmetric spaces. As a byproduct we obtain the Hausdorff dimension of the set of rays with a fixed maximal singular direction, which move away into one end of a locally symmetric space at linear depth, infinitely many times.

In this paper, we extend the theory of simultaneous Diophantine approximation to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very general framework and define what it means for such a theorem to be optimal. We show that optimality is implied by but does not imply the existence of badly approximable points.

Given a real vector =(1; : : : ; d) and a real number " > 0 a good Diophantine approximation to is a number Q such that kQQ mod Zk1 ", where k k1 denotes thè1-norm kxk1 := max 1id jxij for x = (x1; : : : ; x d). Lagarias 12] proved the NP-completeness of the corresponding decision problem, i.e., given a vector 2 Q d , a rational number " > 0 and a number N 2 N+, decide whether there exists a nu...