Metric results on summatory arithmetic functions on Beatty sets

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 Radziwiłł 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.