ON THE DIMENSIONLESSNESS OF INVARIANT SETS

Let M be a subset of R with the following two invariance properties: (1) M+k(subset) M for all integers k, and (2) there exists a positive integer l => 2 such that 1/l M(subset) M. (For example, the set of Liouville numbers and the Besicovitch-Eggleston set of non-normal numbers satisfy these conditions.) We prove that if h is a dimension function that is strongly concave at 0, then the h-dimensional Hausdorff measure H^h(M) of M equals 0 or infinity.