  The goal of this expository article is a fairly self-contained account of
some averaging processes of functions along sequences of the form $(\alpha^n
x)^{}_{n\in\mathbb{N}}$, where $\alpha$ is a fixed real number with $| \alpha |
> 1$ and $x\in\mathbb{R}$ is arbitrary. Such sequences appear in a multitude of
situations including the spectral theory of inflation systems in aperiodic
order. Due to the connection with uniform distribution theory, the results will
mostly be metric in nature, which means that they hold for Lebesgue-almost
every $x\in\mathbb{R}$.
