$q$-Quasiadditive Functions

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$f(q^{k+r}a + b) = f(a) f(b)$forall$b < q^k$andafixedparameter$r$\mathrm{.}Inadditiontosomeelementarypropertiesof$q$-quasiadditiveand$q$-quasimultiplicativefunctions,weprovecharacterisationsof$q$-quasiadditivityand$q$-quasimultiplicativityforthespecialclassof$q$-regular functions. The final main result provides a general central limit theorem that includes both classical and new examples as corollaries.