On a Problem of Schweiger Concerning Normal Numbers

Abstract

Let T and S be two number theoretical transformations on the n-dimensional unit cube B, and write TtS if there exist positive integers m and n such that T m =S n . F. Schweiger showed in [1969, J. Number Theory 1, 390 397] that TtS implies that every T-normal number x is S-normal. Furthermore, he conjectured that T t % S implies that not all T-normal x are S-normal. In this note two counterexamples to this conjecture are given. Academic Press

Generic point equivalence and Pisot numbers

“…The following examples were shown by Kraaikamp and Nakada in [13]. Let β be a Perron number: an algebraic integer greater than one whose conjugates have modulus less than β. Handelman [5] showed that β has no other positive conjugates if and only if there exist an ℓ ∈ N and a nonnegative integer vector (a 1 , .…”

“…Schweiger [22] and Vandehey [24] showed that if two number theoretic transformations T and S satisfy T m = S n for some positive integers n, m, then every T -normality is equivalent to S-normality. Kraaikamp and Nakada [13] gave counter examples that the other direction does not hold. They used the jump transformation to show the equivalence of normality: normality equivalence, in short.…”

Normal equivalencies for eventually periodic basic sequences

“…Vandehey also showed the equivalence of normality with respect to the regular continued fraction expansion and with respect to the odd continued fraction expansion. C. Kraaikamp and H. Nakada [5] answered a conjecture of F. Schweiger [13] on normal numbers. The Q-Cantor series expansions, first studied by G. Cantor in [1], are a natural generalization of the b-ary expansions.…”

“…Theorem 1.4 (C. Kraaikamp and H. Nakada, 2001). There exist number-theoretic transformations S and T such that N(S) = N(T ) and such that there are no positive integers m and n with S m = T n .…”

On the Borel complexity of continued fraction normal, absolutely abnormal numbers

“…So, for instance, it is known that N b = N b k for all b ≥ 2, k ∈ N [28], and it is known that N CF is the same as the set of normal numbers for certain continued fraction variants [23,37]. Some other examples of different digital systems with the same set of normal numbers can be found in [24,34]. On the other hand, few examples of systems with different sets of normal numbers are known.…”