On a problem of Schweiger concerning normal numbers

Let T and S be two number theoretical transformations n-dimensional unit cu be B, and write T similar to S if there exist positive integers m and n suc h that T-m = S-n. F. Schweiger showed in [1969, J. Number, Theory,1, 390-39 7] that T similar toS implies that every T-normal number x is S-normal. Fur thermore, lie conjectured that T similar to S implies that not all T-normal x are S-normal. In this note two counterexamples to this conjecture are gi ven. (C) 2001 Academic Press.