For a positive integer n and a real number ξ, let λ_n(ξ) denote the supremum of the real numbers λ for which there are arbitrarily large positive integers q such that ||qξ||, ||qξ²||, ..., ||qξⁿ|| are all less than q^-λ. Here, || || denotes the distance to the nearest integer. We establish new results on the Hausdorff dimension of the set of real numbers ξ such that λ_n(ξ) is equal (or greater than or equal) to a given value.

The main part of this work has been done while Yann Bugeaud was visiting the University of Sydney, supported by the Sydney Mathematical Research Institute International Visitor Program. The authors are grateful to the referee for a very careful reading.