On inhomogeneous diophantine approximation and Hausdorff dimension

Abstract

Let Γ = ZA + Z n ⊂ R n be a dense subgroup of rank n + 1 and let ŵ(A) denote the exponent of uniform simultaneous rational approximation to the generating point A. For any real number v ≥ ŵ(A), the Hausdorff dimension of the set B v of points in R n that are v-approximable with respect to Γ is shown to be equal to 1/v. © 2012 Springer Science+Business Media, Inc.