The carry propagation of the successor function

Abstract: Given any numeration system, the carry propagation at an integer $N$ is the number of digits that change between the representation of $N$ and $N+1$. The carry propagation of the numeration system as a whole is the average carry propagations at the first $N$ integers, as $N$ tends to infinity, if this limit exists.

In the case of the usual base $p$ numeration system, it can be shown that the limit indeed exists and is equal to $p/(p-1)$. We recover a similar value for those numeration systems we consider and for which the limit exists.

The problem is less the computation of the carry propagation than the proof of its existence. We address it for various kinds of numeration systems: abstract numeration systems, rational base numeration systems, greedy numeration systems and beta-numeration. This problem is tackled with three different types of techniques: combinatorial, algebraic, and ergodic, each of them being relevant for different kinds of numeration systems.

This work has been published in Advances in Applied Mathematics 120 (2020). In this talk, we shall focus on the algebraic and ergodic methods.

Joint work with V. Berthé (Irif), Ch. Frougny (Irif), and M. Rigo (Univ. Liège).

dynamical systemsnumber theory

Audience: researchers in the topic

( paper )

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