Inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems

Abstract

In this paper, we investigate inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems. For β>1 let Tβ be the β-transformation on [0,1]. We determine the Lebesgue measure and Hausdorff dimension of the set{(x,y)∈[0,1]2:|Tβnx−f(x,y)|<φ(n) for infinitely many n∈N}, where f:[0,1]2→[0,1] is a Lipschitz function and φ is a positive function on N. Let β2≥β1>1, f1,f2:[0,1]→[0,1] be two Lipschitz functions, τ1,τ2 be two positive continuous functions on [0,1]. We also determine the Hausdorff dimension of the set{(x,y)∈[0,1]2:|Tβ1nx−f1(x)|<β1−nτ1(x)|Tβ2ny−f2(y)|<β2−nτ2(y) for infinitely many n∈N}. Under certain additional assumptions, the Hausdorff dimension of the set{(x,y)∈[0,1]2:|Tβ1nx−g1(x,y)|<β1−nτ1(x)|Tβ2ny−g2(x,y)|<β2−nτ2(y) for infinitely many n∈N} is also determined, where g1,g2:[0,1]2→[0,1] are two Lipschitz functions.