$q$-Quasiadditive Functions

Abstract

In this paper, we introduce the notion of $q$-quasiadditivity of arithmetic functions, as well as the related concept of $q$-quasimultiplicativity, which generalises strong $q$-additivity and -multiplicativity, respectively. We show that there are many natural examples for these concepts, which are characterised by functional equations of the form $f(q^{k+r}a + b) = f(a) + f(b)$ or $f(q^{k+r}a + b) = f(a) f(b)$ for all $b < q^k$ and a fixed parameter $r$. In addition to some elementary properties of $q$-quasiadditive and $q$-quasimultiplicative functions, we prove characterisations of $q$-quasiadditivity and $q$-quasimultiplicativity for the special class of $q$-regular functions. The final main result provides a general central limit theorem that includes both classical and new examples as corollaries.

Extended Abstract for the 27th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms