Let $f: \mathbb R \rightarrow \mathbb R$ be integrable and $a>0$.
I would like to prove that
$\lim_{n \rightarrow \infty} f(nx) / n^{-a} = 0$ for almost all $x \in \mathbb R$.
We already proved that for almost all $x \in \mathbb R$ the series $\sum_{n \in \mathbb Z} f(n+x/a)$ converges and that
$\int_0^a g(x) dx = a \int_{-\infty}^\infty f(x) dx$ holds where
$g(x) := \sum_{n \in \mathbb Z} f(n+x/a)$, if the series converges, and $0$ else.
Does anybody have an idea how to do that? (I am not sure if the last two statements could be useful)
Thanks in advance