On a Problem of Schweiger Concerning Normal Numbers Original Research Article

Let T and S be two number theoretical transformations on the n-dimensional unit cube B, and write Tnot, vert, similarS if there exist positive integers m and n such that Tm=Sn. F. Schweiger showed in [1969, J. Number Theory1, 390–397] that Tnot, vert, similarS implies that every T-normal number x is S-normal. Furthermore, he conjectured that Tnot similarS implies that not all T-normal x are S-normal. In this note two counterexamples to this conjecture are given.