On q-Quasiadditive and q-Quasimultiplicative Functions

CITATION: Kropf, S. & Wagner, S. 2017. On q-Quasiadditive and q-Quasimultiplicative Functions. Electronic Journal of Combinatorics, 24(1):1-22.The original publication is available at https://www.combinatorics.org/ojs/index.php/eljcIn this paper, we introduce the notion of q-quasiadditivity of arithmetic functions, as well as the related concept of q-quasimultiplicativity, which generalise 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(qk+ra+b) = f(a)+f(b) or f(qk+ra+b) = f(a)f(b) for all b < qk 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.https://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i1p60Publisher's versio

On q-Quasiadditive and q-Quasimultiplicative Functions

CITATION: Kropf, S. & Wagner, S. 2017. On q-Quasiadditive and q-Quasimultiplicative Functions. Electronic Journal of Combinatorics, 24(1):1-22.The original publication is available at https://www.combinatorics.org/ojs/index.php/eljcIn this paper, we introduce the notion of q-quasiadditivity of arithmetic functions, as well as the related concept of q-quasimultiplicativity, which generalise 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(qk+ra+b) = f(a)+f(b) or f(qk+ra+b) = f(a)f(b) for all b < qk 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.https://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i1p60Publisher's versio

On uniformity of qq-multiplicative sequences

We show that any $q$-multiplicative sequence which is \emph{oscillating} of order $1$, i.e.\ does not correlate with linear phase functions $e^{2\pi i n\alpha}$ ($\alpha \in \mathbb{R})$, is Gowers uniform of all orders, and hence in particular does not correlate with polynomial phase functions $e^{2\pi i p(n)}$ ($p \in \mathbb{R}[x]$). Quantitatively, we show that any $q$-multiplicative sequence which is of Gelfond type of order 1 is automatically of Gelfond type of all orders. Consequently, any such $q$-multiplicative sequence is a good weight for ergodic theorems. We also obtain combinatorial corollaries concerning linear patterns in sets which are described in terms of sums of digits.Comment: 25 page