Odometers and systems of numeration. (English) Zbl 0822.11008

Grabner, Peter J.; Liardet, Pierre; Tichy, Robert F.

Acta Arith. 70, No. 2, 103-123 (1995).

The \(G\)-odometer (also called \(G\)-adic machine), where the sequence \(G\) denotes a numeration scale, is a generalization of the addition of 1 in the group of the \(q\)-adic integers (when \(q=2\), the dyadic odometer is isomorphic to the von Neumann-Kakutani’s transformation, also called van der Corput’s transformation). This notion (due to the second author) is very closely connected to the adding shift introduced by Vershik.
This paper establishes dynamical properties of the \(G\)-odometer and, in particular, a necessary and sufficient condition for the continuity. Moreover, when the scale \(G\) satisfies a linear recurrence, the \(G\)- odometer is continuous if and only if the recurrence is finite. Furthermore, the authors prove in this case that under a certain combinatorial hypothesis the odometer has purely discrete spectrum. This property has been proved by Solomyak for linear recurrences with decreasing coefficients.
The last part of this paper states the well-distribution of the sequence \(x s_ G (n)+ yn\), if at least one of the numbers \(x\) and \(y\) is irrational, where \(s_ G(n)\) denotes the sum-of-digits function with respect to the linear recurrence \(G\). This paper ends with a very complete list of references.

Reviewer: V.Berthé (Bordeaux)

MSC:

11A67	Other number representations
11B37	Recurrences
11K36	Well-distributed sequences and other variations
11K38	Irregularities of distribution, discrepancy

Keywords:

numeration system; \(G\)-multiplicative functions; discrepancy; \(G\)- odometer; \(G\)-adic machine; numeration scale; linear recurrence; discrete spectrum; sum-of-digits function