Modified shrinking target problem in beta dynamical systems

Abstract

In the paper, we investigate a modified version of the shrinking target problem in beta dynamical systems which is induced by the β-transformation Tβ:x→βx(mod1). The Hausdorff dimension of the lim sup setW(f,φ)={x∈[0,1):|Tβnx−f(x)|<φ(n)for infinitely many n∈N} is determined, where f:[0,1]→[0,1] is a Lipschitz function and φ is a positive function defined on N. The set W(f,φ) can be viewed as a dynamical analogue of the inhomogeneous Diophantine approximation in which the inhomogeneous factor is allowed to vary.