  Let $f\colon\mathbb{N}\rightarrow\mathbb{C}$ be an arithmetic function and
consider the Beatty set $\mathcal{B}(\alpha) = \lbrace\, \lfloor n\alpha
\rfloor : n\in\mathbb{N} \,\rbrace$ associated to a real number $\alpha$, where
$\lfloor\xi\rfloor$ denotes the integer part of a real number $\xi$. We show
that the asymptotic formula \[
  \Bigl\lvert
  \sum_{\substack{ 1\leq m\leq x \\ m\in \mathcal{B}(\alpha) }} f(m)
  - \frac{1}{\alpha} \sum_{1\leq m\leq x} f(m)
  \Bigr\rvert^2
  \ll_{f,\alpha,\varepsilon}
  (\log x)
  (\log\log x)^{3+\varepsilon}
  \sum_{1\leq m\leq x} \lvert f(m) \rvert^2 \] holds for almost all $\alpha>1$
with respect to the Lebesgue measure. This significantly improves an earlier
result due to Abercrombie, Banks, and Shparlinski. The proof uses a recent
Fourier-analytic result of Lewko and Radziwi{\l}{\l} based on the classical
Carleson--Hunt inequality. Moreover, using a probabilistic argument, we
establish the existence of functions $f\colon\mathbb{N}\to\lbrace\,\pm
1\,\rbrace$ for which the above error term is optimal up to logarithmic
factors.
