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.
Categories: math.NT