Additive Functions with Respect to Numeration Systems on Regular Languages

Additive numeration systems and the corresponding additive arithmetic functions have been studied from various points of view since the seminal papers of H. Delange [4, 5], where such functions were investigated for the usual q-adic numeration system. Later more exotic systems of numeration, such as general linear numeration systems [13, 14], especially such systems defined by linear recurring sequences were considered. Furthermore, digital representations with respect to substitutions over a finite alphabet were studied (cf. [8, 9]). All these numeration systems have in common that the number of integers represented by words of length n satisfies a pure exponential law ∼ C for some constant C > 1. Different aspects of such representations of the integers were studied: dynamics of corresponding adding machine (“odometer”) [15], topological dynamics of the odometer [1], asymptotic properties of summatory functions of additive functions such as the “sumof-digits” function [8, 9, 12, 16, 17], local and global versions of central limit theorems for the values of additive functions [6, 7, 10], existence of distribution functions of additive functions [2]. In the present paper we take the opposite approach compared to the existing literature on the subject. We start with a regular language L, order its words by the genealogical ordering induced by an ordered alphabet, and assign the number n the (n+1)-st word in the language. The above mentioned expansions, which come from finite linear recurrences are special cases of this setting. For the question of recognizability of the language generated by an increasing sequence of integers we refer to [13, 20, 21]. The paper is organized as follows. In Section 2 we introduce the basic notation of numeration systems related to regular languages; for this purpose we summarize the contents of the paper [18]. In Section 3 we state the main theorem, which is the appropriate analogue of the summation formula for the sum-of-digits function discovered by H. Delange