Hausdorff Dimension in Inhomogeneous Diophantine Approximation

Abstract

Let alpha be an irrational real number. We show that the set of epsilon-badly approximable numbers Bad(epsilon) (alpha) := {x is an element of [0, 1] : lim inf(vertical bar|q vertical bar ->infinity vertical bar)q vertical bar . parallel to q alpha - x parallel to >= epsilon} has full Hausdorff dimension for some positive epsilon if and only if alpha is singular on average. The condition is equivalent to the average 1/k Sigma(i=1, ...,k) log a(i) of the logarithms of the partial quotients a(i) of alpha going to infinity with k. We also consider one-sided approximation, obtain a stronger result when a(i) tends to infinity, and establish a partial result in higher dimensions.