Inhomogeneous Diophantine approximation with general error functions

Lingmin Liao (1) , Michal Rams (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 PAN - Institut of Mathematics - Polish Academy of Sciences

Let $\al$ be an irrational and $\varphi: \N \rightarrow \R^+$ be a function decreasing to zero. For any $\al$ with a given Diophantine type, we show some sharp estimations for the Hausdorff dimension of the set \[ E_{\varphi}(\al):=\{y\in \R: \|n\al -y\| < \varphi(n) \text{ for infinitely many } n\}, \] where $\|\cdot\|$ denotes the distance to the nearest integer.

Mots clés: Inhomogeneous Diophantine approximation; Hausdorff dimension

Domaines: Systèmes dynamiques [math.DS]

Lingmin Liao, Michal Rams. Inhomogeneous Diophantine approximation with general error functions. Acta Arithmetica, 2013, 160 (1), pp.25--35. ⟨hal-00723320v3⟩