HAUSDORFF DIMENSION OF THE SET APPROXIMATED BY IRRATIONAL ROTATIONS

Abstract

Let theta be an irrational number and phi : N -> R+ be a monotone decreasing function tending to zero. Let E-phi(theta) = {y is an element of R : parallel to n theta - y parallel to < phi(n), for infinitely many n is an element of N}, i.e. the et of points which are approximated by the irrational rotation with respect to the error function phi(n). In this article, we give a complete description of the Hausdorff dimension of E-phi(theta) for any monotone function phi and any irrational theta.