7 research outputs found
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 -multiplicative sequences
We show that any -multiplicative sequence which is \emph{oscillating} of
order , i.e.\ does not correlate with linear phase functions (, is Gowers uniform of all orders, and hence
in particular does not correlate with polynomial phase functions (). Quantitatively, we show that any
-multiplicative sequence which is of Gelfond type of order 1 is
automatically of Gelfond type of all orders. Consequently, any such
-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