On the joint normality of certain digit expansions

Abstract

We prove that a point x is normal with respect to an ergodic, numbertheoretic transformation T if and only if x is normal with respect to T n for any n ≥ 1. This corrects an erroneous proof of Schweiger. Then, using some insights from Schweiger's original proof, we extend these results, showing for example that a number is normal with respect to the regular continued fraction expansion if and only if it is normal with respect to the odd continued fraction expansion.

Preservation of normality by non-oblivious group selection

“…This is a simplified verison of Theorem 3.1 in [18] (see Remark 3.2 in that paper for the discussion of the necessary conditions needed on the dynamic system and note that they are all trivial in our case). See also [17] for a simpler proof.…”

Towards a sharp converse of Wall’s theorem on arithmetic progressions

“…Schweiger [7] was the first to state this result, although his proof in one direction was erroneous. See [9] for a corrected proof. In the special case of base-b expansions, this was found several years earlier.…”

Normal equivalencies for eventually periodic basic sequences

“…Generalizations of Theorem 1.1 have been considered for T -normal numbers. J. Vandehey [16] presented a corrected proof of a result announced by F. Schweiger in [13].…”