Metric results on summatory arithmetic functions on Beatty sets

Abstract

Let $f:\mathbb{N}\to \mathbb{C}$ be an arithmetic function and consider the Beatty set $\mathcal{B}(\hat{I}\pm )=\{\lfloor n\hat{I}\pm \rfloor :n\in \mathbb{N}\}$ associated to a real number $\hat{I}\pm$, where $\lfloor \hat{I}¾\rfloor$ denotes the integer part of a real number $\hat{I}¾$. We show that the asymptotic formula \[ \Bigl\lvert \sum_{\substack{ 1\leq m\leq x \\ m\in \mathcal{B}(α) }} f(m) - \frac{1}α \sum_{1\leq m\leq x} f(m) \Bigr\rvert^2 \ll_{f,α,\varepsilon} (\log x) (\log\log x)^{3+\varepsilon} \sum_{1\leq m\leq x} \lvert f(m) \rvert^2 \] holds for almost all $\hat{I}\pm >1$ 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 $f:\mathbb{N}\to \{\pm 1\}$ for which the above error term is optimal up to logarithmic factors.

11 pages